Q.Integrate the function $\frac{\sin^8 x-\cos^8 x}{1-2\sin^2 x\cos^2 x}$

Showing the 12 most recent of 34 on this concept.

1. TG EAPCET 2026Set eng-2026-05-10-FN1 markMCQ
   Q.If $y = \sin^{-1} \left( \frac{1 - \cos 2x}{1 + \sin^4 x} \right)$ then $\frac{dy}{dx} =$ (A) $\frac{2 \cos 2x}{1 + \sin^4 x}$ (B) $\frac{2 \sin 2x}{1 + \sin^4 x}$ (C) $\frac{2 \cos 2x}{1 + \sin^8 x}$ (D) $\frac{2 \sin 2x}{1 + \sin^8 x}$

   ›Reveal solutionSolution

   The key is to simplify the argument of the inverse sine using trigonometric identities before differentiating. After simplification, the derivative becomes $\frac{2 \sin 2x}{1 + \sin^4 x}$, which matches option (B).

   ---

   We start with

   $y = \sin^{-1}\left( \frac{1 - \cos 2x}{1 + \sin^4 x} \right).$

   Concept and intuition

   The expression inside the inverse sine looks messy, but the numerator $1 - \cos 2x$ is a classic form that simplifies to $2\sin^2 x$. That’s a huge clue: the whole fraction might simplify to something like $\frac{2\sin^2 x}{1 + \sin^4 x}$, which is a perfect candidate for the identity $\sin^{-1}\left(\frac{2t}{1+t^2}\right) = 2\tan^{-1} t$ (for $|t|\le 1$). That substitution makes differentiation straightforward.

   ---

   Step-by-step

   1. Simplify the numerator Using the double-angle identity:

   $1 - \cos 2x = 2\sin^2 x.$

   So the argument becomes

   $\frac{2\sin^2 x}{1 + \sin^4 x}.$

   2. Recognize a standard form Let $t = \sin^2 x$. Then the argument is

   $\frac{2t}{1 + t^2}.$

   This is exactly the form $\frac{2t}{1+t^2}$, which appears in the identity

   $\sin^{-1}\left( \frac{2t}{1+t^2} \right) = 2\tan^{-1} t \quad \text{for } |t| \le 1.$

   Since $t = \sin^2 x \in [0,1]$, the condition holds. Therefore

   $y = 2\tan^{-1}(\sin^2 x).$

   3. Differentiate Differentiate $y = 2\tan^{-1}(\sin^2 x)$:

   $\frac{dy}{dx} = 2 \cdot \frac{1}{1 + (\sin^2 x)^2} \cdot \frac{d}{dx}(\sin^2 x).$

   The derivative of $\sin^2 x$ is $2\sin x \cos x = \sin 2x$. So …

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2. TG EAPCET 2023Set eng-2023-05-12-FN1 markMCQ
   Q.If $\tan y = \cot \left( \frac{\pi}{4} - x \right)$ then $\frac{dy}{dx} =$ (A) $\frac{\csc^2 \left( \frac{\pi}{4} - x \right)}{1 + \cot^2 \left( \frac{\pi}{4} + x \right)}$ (B) $\frac{-\csc^2 \left( \frac{\pi}{4} - x \right)}{\sec^2 y}$ (C) $\frac{\csc^2 \left( \frac{\pi}{4} - x \right)}{1 + \tan^2 \left( \frac{\pi}{4} - x \right)}$ (D) $\frac{\sec^2 \left( \frac{\pi}{4} + x \right)}{1 + \tan^2 \left( \frac{\pi}{4} + x \right)}$

   ›Reveal solutionSolution

   Differentiate $\tan y=\cot(\tfrac\pi4-x)$ implicitly: $\sec^2y\,\dfrac{dy}{dx}=\csc^2(\tfrac\pi4-x)$. Since $\csc^2(\tfrac\pi4-x)=\sec^2(\tfrac\pi4+x)=\sec^2y$, the derivative is $1$, and option (D) is the form equal to $1$.

   1. Simplify the relation. Using $\cot\theta=\tan(\tfrac\pi2-\theta)$,

   $\cot\!\left(\frac\pi4-x\right)=\tan\!\left(\frac\pi2-\frac\pi4+x\right)=\tan\!\left(\frac\pi4+x\right),$

   so $\tan y=\tan(\tfrac\pi4+x)$, i.e. $y=\tfrac\pi4+x+n\pi$ and $\sec^2y=\sec^2(\tfrac\pi4+x)$.

   2. Differentiate implicitly. With $u=\tfrac\pi4-x$, $u'=-1$:

   $\sec^2y\,\frac{dy}{dx}=-\csc^2\!\left(\frac\pi4-x\right)\cdot(-1)=\csc^2\!\left(\frac\pi4-x\right).$

   3. Convert with a cofunction identity. Because $\sin(\tfrac\pi4-x)=\cos(\tfrac\pi4+x)$,

   $\csc^2\!\left(\frac\pi4-x\right)=\sec^2\!\left(\frac\pi4+x\right).$

   Therefore …

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3. TG EAPCET 2026Set eng-2026-05-10-AN1 markMCQ
   Q.If $x = \sin^{-1}(\cos t)$ and $y = \tan^{-1}(\cos t)$, then $\dfrac{dy}{dx} =$ (A) $\dfrac{\cos x}{1 + \sin^2 x}$ (B) $\dfrac{\cos t}{1 + \sin^2 t}$ (C) $\dfrac{\sin t}{1 + \sin^2 t}$ (D) $\dfrac{\sin x}{1 + \cos^2 x}$

   ›Reveal solutionSolution

   The key is to express both $x$ and $y$ in terms of $t$, then use the chain rule to find $\frac{dy}{dx}$ as a function of $x$, matching option (A).

   We are given:

   $x = \sin^{-1}(\cos t), \quad y = \tan^{-1}(\cos t).$

   We need $\frac{dy}{dx}$.

   Concept and intuition:

   Both $x$ and $y$ are functions of $t$, so we can compute $\frac{dy}{dt}$ and $\frac{dx}{dt}$ separately, then use $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$. The trick is to simplify $\sin^{-1}(\cos t)$ into a nicer algebraic form in terms of $t$, which will make differentiation cleaner. Then we express the final answer in terms of $x$ (since the options involve $x$ or $t$).

   ---

   1. Simplify $x$ Recall: $\sin^{-1}(\cos t) = \frac{\pi}{2} - t$ for $t$ in $[0, \pi]$, but more generally we can use the identity $\sin^{-1}(\cos t) = \frac{\pi}{2} - |t|$? Actually, a cleaner approach: Let $u = \cos t$. Then $x = \sin^{-1}(u)$. But we can also write $\cos t = \sin\left(\frac{\pi}{2} - t\right)$, so

   $x = \sin^{-1}\left(\sin\left(\frac{\pi}{2} - t\right)\right).$

   For the principal range of $\sin^{-1}$, we need $\frac{\pi}{2} - t \in [-\pi/2, \pi/2]$, i.e., $t \in [0, \pi]$. Assuming this, we get

   $x = \frac{\pi}{2} - t.$

   Hence, $t = \frac{\pi}{2} - x$.

   2. Differentiate $x$ and $y$ with respect to $t$ From $x = \frac{\pi}{2} - t$, we have

   $\frac{dx}{dt} = -1.$

   For $y = \tan^{-1}(\cos t)$, differentiate:

   $\frac{dy}{dt} = \frac{1}{1 + (\cos t)^2} \cdot (-\sin t) = -\frac{\sin t}{1 + \cos^2 t}.$

   3. Find $\frac{dy}{dx}$ Using the chain rule: $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{-\frac{\sin t}{1 + \cos^2 t}}{-1} = \frac{\sin t}{1 + \cos^2 t}.$ …

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4. TG EAPCET 2025Set eng-2025-05-04-AN1 markMCQ
   Q.$\lim_{x \to 0} \frac{\sqrt{\cos x} - \sqrt[3]{\cos x}}{\sin^2 x} =$ (A) $\frac{1}{2}$ (B) $-\frac{1}{12}$ (C) $-\frac{1}{6}$ (D) $\frac{2}{3}$

   ›Reveal solutionSolution

   The limit simplifies by factoring out $\sqrt[3]{\cos x}$ and using series expansions for $\cos x$ and the binomial expansion; the result is $-\frac{1}{12}$, so the correct option is (B).

   Concept & Intuition

   When $x \to 0$, both numerator and denominator approach $0$, giving a $\frac{0}{0}$ form. Direct substitution fails. The trick: rewrite the numerator as a common power of $\cos x$, then expand $\cos x$ as a series near $0$: $\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} + \cdots$. The square and cube roots become binomial expansions $(1+u)^p \approx 1 + pu + \frac{p(p-1)}{2}u^2 + \cdots$, which lets us isolate the leading-order cancellation. The denominator $\sin^2 x \approx x^2$ sets the scale.

   Step-by-step solution

   1. Rewrite the numerator with a common factor Let $t = \cos x$. Then the numerator is $t^{1/2} - t^{1/3}$. Factor out $t^{1/3}$:

   $\sqrt{\cos x} - \sqrt[3]{\cos x} = (\cos x)^{1/3}\left[(\cos x)^{1/6} - 1\right].$

   As $x \to 0$, $\cos x \to 1$, so $(\cos x)^{1/3} \to 1$ and won't affect the limit's value. The key is the bracket $(\cos x)^{1/6} - 1$.

   2. Expand $\cos x$ near $0$

   $\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} + O(x^6).$

   Let $u = -\frac{x^2}{2} + \frac{x^4}{24} + \cdots$, so $\cos x = 1 + u$ with $u \to 0$.

   3. Binomial expansion for $(1+u)^{1/6}$

   $(1+u)^{1/6} = 1 + \frac{1}{6}u + \frac{(1/6)(1/6 - 1)}{2}u^2 + O(u^3).$

   Compute $u$ and $u^2$ to order $x^4$:

   • $u = -\frac{x^2}{2} + \frac{x^4}{24} + \cdots$
   • $u^2 = \left(-\frac{x^2}{2}\right)^2 + \cdots = \frac{x^4}{4} + \cdots$ (higher terms are $O(x^6)$).

   Then

   $(1+u)^{1/6} = 1 + \frac{1}{6}\left(-\frac{x^2}{2} + \frac{x^4}{24}\right) + \frac{(1/6)(-5/6)}{2}\cdot\frac{x^4}{4} + O(x^6).$

   Simplify:

   • Linear term: $-\frac{x^2}{12} + \frac{x^4}{144}$
   • Quadratic term: $\frac{-5/36}{2} \cdot \frac{x^4}{4} = -\frac{5}{72} \cdot \frac{x^4}{4} = -\frac{5x^4}{288}$.

   So

   $(\cos x)^{1/6} = 1 - \frac{x^2}{12} + \left(\frac{1}{144} - \frac{5}{288}\right)x^4 + O(x^6) = 1 - \frac{x^2}{12} - \frac{3}{288}x^4 + \cdots = 1 - \frac{x^2}{12} - \frac{x^4}{96} + \cdots.$

   4. Thus the bracket

   $(\cos x)^{1/6} - 1 = -\frac{x^2}{12} - \frac{x^4}{96} + O(x^6).$

   5. Denominator expansion $$\sin^2 x = \left(x - \frac{x^3}{6} + \cdots\right)^2 = x^2 - \frac{x^4}{3} + O(x^6). …$$

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5. TG EAPCET 2021Set eng-2021-08-04-AN1 markMCQ
   Q.$\lim_{x \to 0} \frac{x^2 2^x - x^2 \sin x - x^2}{3^x + \cos x - 3^x \cos x - 1} =$ (A) $\frac{1}{\log 3} (\log 2 - 1)$ (B) $\frac{4}{\log 3} (1 - \log 2)$ (C) $\frac{4}{\log 3} (\log 2 - 1)$ (D) $\frac{2}{\log 3} (\log 2 - 1)$

   ›Reveal solutionSolution

   Factor the denominator as $(1-\cos x)(3^x-1)$ and expand the numerator to leading order $x^3(\log2-1)$; the limit is $\dfrac{2}{\log3}(\log2-1)$ — option (D).

   Factor the denominator.

   $3^x+\cos x-3^x\cos x-1=3^x(1-\cos x)-(1-\cos x)=(1-\cos x)(3^x-1).$

   As $x\to0$: $1-\cos x\sim\dfrac{x^2}{2}$ and $3^x-1\sim x\log3$, so the denominator $\sim\dfrac{x^3\log3}{2}$.

   Expand the numerator $x^2\big(2^x-\sin x-1\big)$, using $2^x=1+x\log2+O(x^2)$ and $\sin x=x+O(x^3)$: …

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6. TG EAPCET 2025Set eng-2025-05-04-FN1 markMCQ
   Q.$\lim_{x \to 0} \frac{\sqrt{\cos x} - \sqrt[3]{\cos x}}{\sin^2 x} =$ (A) $\frac{2}{3}$ (B) $-\frac{1}{6}$ (C) $-\frac{1}{12}$ (D) $\frac{1}{2}$

   ›Reveal solutionSolution

   The limit simplifies by factoring out a common power of $\cos x$ and using the series expansions for $\cos x$ and $(1+u)^\alpha$; the final value is $-\frac{1}{12}$, which corresponds to option (C).

   We want

   $L = \lim_{x \to 0} \frac{\sqrt{\cos x} - \sqrt[3]{\cos x}}{\sin^2 x}.$

   Both numerator and denominator vanish as $x \to 0$, so this is a $\frac{0}{0}$ form. The key is to rewrite the numerator in terms of a common factor and then use expansions near $x=0$.

   ---

   1. Factor out the smallest power of $\cos x$.

   Write $\sqrt{\cos x} = (\cos x)^{1/2}$ and $\sqrt[3]{\cos x} = (\cos x)^{1/3}$. The smaller exponent is $\frac{1}{3}$, so factor $(\cos x)^{1/3}$ out:

   $(\cos x)^{1/2} - (\cos x)^{1/3} = (\cos x)^{1/3}\left[(\cos x)^{1/6} - 1\right].$

   Thus

   $L = \lim_{x \to 0} \frac{(\cos x)^{1/3}\left[(\cos x)^{1/6} - 1\right]}{\sin^2 x}.$

   As $x \to 0$, $\cos x \to 1$, so $(\cos x)^{1/3} \to 1$. That factor is harmless; the interesting part is the bracket.

   ---

   2. Expand $\cos x$ near $0$.

   We know

   $\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} + O(x^6).$

   Also $\sin^2 x = x^2 - \frac{x^4}{3} + O(x^6)$.

   ---

   3. Expand $(\cos x)^{1/6}$ using $(1+u)^\alpha$.

   Let $u = \cos x - 1 = -\frac{x^2}{2} + \frac{x^4}{24} + \cdots$. Then

   $(\cos x)^{1/6} = (1+u)^{1/6} = 1 + \frac{1}{6}u + \frac{(1/6)(1/6 - 1)}{2}u^2 + \cdots$

   $= 1 + \frac{1}{6}\left(-\frac{x^2}{2} + \frac{x^4}{24}\right) + \frac{1}{6}\left(-\frac{5}{6}\right)\frac{1}{2}\left(-\frac{x^2}{2}\right)^2 + O(x^6).$

   Compute term by term:

   • Linear in $u$: $\frac{1}{6}\left(-\frac{x^2}{2}\right) = -\frac{x^2}{12}$, and the $\frac{x^4}{24}$ part gives $+\frac{x^4}{144}$.
   • Quadratic in $u$: $\frac{1}{6}\cdot\left(-\frac{5}{6}\right)\cdot\frac{1}{2} = -\frac{5}{72}$, times $u^2 = \left(-\frac{x^2}{2}\right)^2 = \frac{x^4}{4}$, gives $-\frac{5}{72}\cdot\frac{x^4}{4} = -\frac{5x^4}{288}$.

   So

   $(\cos x)^{1/6} - 1 = -\frac{x^2}{12} + \left(\frac{1}{144} - \frac{5}{288}\right)x^4 + O(x^6).$

   The $x^4$ coefficient: $\frac{1}{144} = \frac{2}{288}$, so $\frac{2}{288} - \frac{5}{288} = -\frac{3}{288} = -\frac{1}{96}$.

   Thus

   $(\cos x)^{1/6} - 1 = -\frac{x^2}{12} - \frac{x^4}{96} + O(x^6).$

   ---

   4. Assemble the limit. …

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7. TG EAPCET 2021Set eng-2021-08-04-AN1 markMCQ
   Q.$\lim_{x \to 0} \frac{(1 - \cos 2x)}{x \tan 2x + \frac{2x}{3} \tan 3x} =$ (A) $-6$ (B) $\frac{1}{2}$ (C) $0$ (D) $\frac{-6}{5}$

   ›Reveal solutionSolution

   This problem involves evaluating a limit that results in an indeterminate form $0/0$. We resolve this by using the trigonometric identity $1 - \cos 2x = 2 \sin^2 x$ and then applying standard limits for $\sin x / x$ and $\tan x / x$ as $x \to 0$. The final value of the limit is $\boxed{\frac{1}{2}}$.

   When evaluating limits, the first step is always to try direct substitution. If this yields a finite number, that's your limit. However, if it results in an indeterminate form like $0/0$ or $\infty/\infty$, it means the function's behavior near that point is not immediately obvious, and further manipulation is required. For expressions involving trigonometric functions as $x \to 0$, we often rely on a set of fundamental limits.

   The core idea here is to transform the given expression into a form where these standard limits can be directly applied. This usually involves using trigonometric identities to simplify terms and then dividing the numerator and denominator by appropriate powers of $x$ to create terms like $\frac{\sin kx}{kx}$ or $\frac{\tan kx}{kx}$, which approach $1$ as $x \to 0$.

   Let's break down the solution step-by-step.

   1. Check for Indeterminate Form

      First, substitute $x=0$ into the expression:

      Numerator: $1 - \cos(2 \cdot 0) = 1 - \cos(0) = 1 - 1 = 0$.

      Denominator: $0 \cdot \tan(2 \cdot 0) + \frac{2 \cdot 0}{3} \tan(3 \cdot 0) = 0 \cdot \tan(0) + 0 \cdot \tan(0) = 0 \cdot 0 + 0 \cdot 0 = 0$.

      Since we get the form $\frac{0}{0}$, the limit is indeterminate, and we need to simplify the expression.

   2. Apply Trigonometric Identity

      The term $1 - \cos 2x$ in the numerator is a common form that can be simplified using the double-angle identity for cosine: $\cos 2x = 1 - 2 \sin^2 x$.

      Rearranging this, we get:

      $1 - \cos 2x = 2 \sin^2 x$.

      Substituting this into the limit expression:

   $\lim_{x \to 0} \frac{2 \sin^2 x}{x \tan 2x + \frac{2x}{3} \tan 3x}$

   3. Prepare for Standard Limits We know the standard limits:

      $\lim_{x \to 0} \frac{\sin x}{x} = 1$

      $\lim_{x \to 0} \frac{\tan x}{x} = 1$

      To use these, we need to divide the numerator and denominator by an appropriate power of $x$.

      The numerator has $\sin^2 x$, which suggests dividing by $x^2$. The denominator has terms like $x \tan 2x$ and $x \tan 3x$. If we divide by $x^2$, these become $\frac{\tan 2x}{x}$ and $\frac{\tan 3x}{x}$, which are suitable for the standard limit form.

      So, divide both the numerator and the denominator by $x^2$:

   $\lim_{x \to 0} \frac{\frac{2 \sin^2 x}{x^2}}{\frac{x \tan 2x + \frac{2x}{3} \tan 3x}{x^2}}$

   $\lim_{x \to 0} \frac{2 \left(\frac{\sin x}{x}\right)^2}{\frac{\tan 2x}{x} + \frac{2}{3} \frac{\tan 3x}{x}}$

   4. Manipulate Terms to Match Standard Forms Now, let's adjust the terms in the denominator to perfectly match the standard limit form $\frac{\tan kx}{kx}$: For $\frac{\tan 2x}{x}$, multiply and divide by $2$: $\frac{\tan 2x}{x} = \frac{\tan 2x}{x} \cdot \frac{2}{2} = 2 \left(\frac{\tan 2x}{2x}\right)$ …

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8. TG EAPCET 2024Set eng-2024-05-09-AN1 markMCQ
   Q.If $\cos x + \cos y = \dfrac{2}{3}$ and $\sin x - \sin y = \dfrac{3}{4}$, then $\sin(x-y) + \cos(x-y) =$ (A) $\dfrac{161}{145}$ (B) $\dfrac{127}{145}$ (C) $\dfrac{1}{2}$ (D) $\dfrac{8}{9}$

   ›Reveal solutionSolution

   We use sum‑to‑product identities to find $\cos\frac{x+y}{2}$ and $\sin\frac{x+y}{2}$, then compute $\sin(x-y)$ and $\cos(x-y)$ via double‑angle formulas, obtaining $\frac{161}{145}$.

   Concept & Intuition

   We are given two equations mixing sums and differences of sines and cosines. The classic trick is to rewrite each as a product using sum‑to‑product identities. That isolates the half‑sum and half‑difference angles. Then we can find $\sin\frac{x-y}{2}$ and $\cos\frac{x-y}{2}$ from the given numbers, and finally use double‑angle formulas to get $\sin(x-y)$ and $\cos(x-y)$.

   ---

   1. Apply sum‑to‑product identities

   $\cos x + \cos y = 2\cos\frac{x+y}{2}\cos\frac{x-y}{2} = \frac{2}{3}$

   $\sin x - \sin y = 2\cos\frac{x+y}{2}\sin\frac{x-y}{2} = \frac{3}{4}$

   2. Divide the two equations to eliminate $\cos\frac{x+y}{2}$ (provided it is nonzero):

   $\frac{2\cos\frac{x+y}{2}\sin\frac{x-y}{2}}{2\cos\frac{x+y}{2}\cos\frac{x-y}{2}} = \frac{3/4}{2/3}$

   $\tan\frac{x-y}{2} = \frac{3}{4} \cdot \frac{3}{2} = \frac{9}{8}$

   3. Find $\sin\frac{x-y}{2}$ and $\cos\frac{x-y}{2}$ from the tangent. Let $t = \frac{x-y}{2}$. Then $\tan t = \frac{9}{8}$. Construct a right triangle: opposite = 9, adjacent = 8, hypotenuse = $\sqrt{9^2+8^2} = \sqrt{145}$. Hence

   $\sin t = \frac{9}{\sqrt{145}}, \qquad \cos t = \frac{8}{\sqrt{145}}.$

   4. Use double‑angle formulas to get $\sin(x-y)$ and $\cos(x-y)$: $$$$

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9. TG EAPCET 2022Set eng-2022-07-18-FN1 markMCQ
   Q.If $x = \csc \theta - \sin \theta$, $y = \csc^{2022} \theta - \sin^{2022} \theta$ and $\left( \dfrac{dy}{dx} \right)^2 = \dfrac{k(y^2 + 4)}{g(x)}$ where $k \in \mathbb{R}$, then $10 + k - g(2022) =$ (A) 0 (B) 6 (C) 10 (D) 14

   ›Reveal solutionSolution

   Using $\csc\theta\cdot\sin\theta=1$ one gets the standard identity $\left(\dfrac{dy}{dx}\right)^2=\dfrac{n^2\,(y^2+4)}{x^2+4}$ with $n=2022$, so $k=2022^2$ and $g(x)=x^2+4$; then $10+k-g(2022)=6$, option (B).

   Write $c=\csc\theta$ and $s=\sin\theta$, noting the key relation $cs=\csc\theta\sin\theta=1$.

   Setting up $x^2+4$ and $y^2+4$. With $x=c-s$ and $n=2022$, $y=c^{n}-s^{n}$:

   $x^2+4=(c-s)^2+4=c^2+s^2-2cs+4=c^2+s^2+2=(c+s)^2,$

   $y^2+4=(c^{n}-s^{n})^2+4=c^{2n}+s^{2n}-2(cs)^n+4=c^{2n}+s^{2n}+2=(c^{n}+s^{n})^2,$

   where $cs=1$ was used so $2cs=2$ and $(cs)^n=1$.

   The derivative. Differentiating with respect to $\theta$ and forming $\dfrac{dy}{dx}=\dfrac{dy/d\theta}{dx/d\theta}$, the algebra collapses to the compact identity

   $\left(\frac{dy}{dx}\right)^2=\frac{n^2\,(y^2+4)}{x^2+4}.$ …

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10. TG EAPCET 2021Set eng-2021-08-06-AN1 markMCQ
    Q.If $\theta$ is the acute angle between the curves $x^2 + y^2 = 2020\sqrt{2}$ and $x^2 - y^2 = 2020$, then

    [!FORMULA] $\frac{\sin \theta + \cos \theta}{\tan \theta} =$

    (A) $\sqrt{2}$ (B) $\frac{3 + \sqrt{3}}{2}$ (C) $\frac{3 + \sqrt{3}}{4}$ (D) $\frac{3 + \sqrt{3}}{6}$

    ›Reveal solutionSolution

    The curves cut at $45^\circ$, so $\dfrac{\sin\theta+\cos\theta}{\tan\theta}=\dfrac{\sqrt2}{1}=\sqrt2$.

    Slopes at a point of intersection.

    Circle $x^2+y^2=2020\sqrt2$: $\;2x+2y\,y'=0\Rightarrow m_1=-\dfrac{x}{y}$.

    Hyperbola $x^2-y^2=2020$: $\;2x-2y\,y'=0\Rightarrow m_2=\dfrac{x}{y}$.

    $\tan\theta=\left|\frac{m_1-m_2}{1+m_1m_2}\right| =\left|\frac{-2x/y}{1-x^2/y^2}\right| =\left|\frac{-2xy}{y^2-x^2}\right|.$

    Point of intersection. Subtracting the equations, $2y^2=2020(\sqrt2-1)$ and $2x^2=2020(\sqrt2+1)$, so

    $y^2-x^2=1010\big[(\sqrt2-1)-(\sqrt2+1)\big]=-2020,$

    $x^2y^2=1010^2(\sqrt2+1)(\sqrt2-1)=1010^2\Rightarrow |xy|=1010.$

    Hence …

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11. TG EAPCET 2023Set eng-2023-05-14-FN1 markMCQ
    Q.If $x = \log(y + \sqrt{y^2 + 1})$ then $y =$ (A) $\tanh x$ (B) $\coth x$ (C) $\sinh x$ (D) $\cosh x$

    ›Reveal solutionSolution

    The expression $x = \log(y + \sqrt{y^2+1})$ is the definition of the inverse hyperbolic sine, so $y = \sinh x$. The correct option is (C).

    The key here is recognizing a standard identity. The expression $y + \sqrt{y^2+1}$ looks like something that appears when you solve for $y$ in terms of $x$ from the definition of hyperbolic sine. In fact, $\sinh x = \frac{e^x - e^{-x}}{2}$, and its inverse is exactly $\sinh^{-1} y = \log(y + \sqrt{y^2+1})$. So the problem is simply asking: if $x = \sinh^{-1} y$, what is $y$? The answer is $y = \sinh x$.

    Let’s verify this step by step.

    1. Start with the given equation

       We have $x = \log(y + \sqrt{y^2 + 1})$. This is an equation relating $x$ and $y$. Our goal is to solve for $y$ in terms of $x$.

    2. Exponentiate both sides

       Since the logarithm is natural log (base $e$), we write

    $e^x = y + \sqrt{y^2 + 1}.$

    This removes the log and gives a simpler equation.

    3. Consider the conjugate expression A classic trick: if $e^x = y + \sqrt{y^2+1}$, then its reciprocal is

    $e^{-x} = \frac{1}{y + \sqrt{y^2+1}}.$

    Rationalize the denominator:

    $e^{-x} = \frac{y - \sqrt{y^2+1}}{(y + \sqrt{y^2+1})(y - \sqrt{y^2+1})} = \frac{y - \sqrt{y^2+1}}{y^2 - (y^2+1)} = \frac{y - \sqrt{y^2+1}}{-1} = \sqrt{y^2+1} - y.$

    So we have two equations:

    $e^x = y + \sqrt{y^2+1}, \quad e^{-x} = \sqrt{y^2+1} - y.$

    4. Subtract to isolate $y$ Subtract the second equation from the first: $$$$

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12. TG EAPCET 2021Set eng-2021-08-06-FN1 markMCQ
    Q.$2 \tan^{-1} \left( \dfrac{1}{3} \right) + \tan^{-1} \left( \dfrac{1}{7} \right) =$ (A) $\tan^{-1} \left( \dfrac{49}{29} \right)$ (B) $\dfrac{\pi}{2}$ (C) $0$ (D) $\dfrac{\pi}{4}$

    ›Reveal solutionSolution

    We simplify the expression by first converting $2 \tan^{-1} \left( \frac{1}{3} \right)$ into a single $\tan^{-1}$ term, then combining it with $\tan^{-1} \left( \frac{1}{7} \right)$ using the sum formula for inverse tangents. The final result is $\boxed{\frac{\pi}{4}}$.

    The problem asks us to evaluate an expression involving inverse tangent functions. The key to solving this is to use the standard addition formulas for inverse tangents to simplify the expression step-by-step. We have a term of the form $2 \tan^{-1} x$ and then a sum of two $\tan^{-1}$ terms.

    Here are the relevant formulas we will use:

    $2 \tan^{-1} x = \tan^{-1} \left( \frac{2x}{1-x^2} \right)$, for $-1 < x < 1$.

    [!FORMULA]

    $\tan^{-1} x + \tan^{-1} y = \tan^{-1} \left( \frac{x+y}{1-xy} \right)$, for $xy < 1$.

    Let's break down the calculation.

    1. Simplify the $2 \tan^{-1} \left( \frac{1}{3} \right)$ term: We start by simplifying the first part of the expression, $2 \tan^{-1} \left( \frac{1}{3} \right)$. We use the formula $2 \tan^{-1} x = \tan^{-1} \left( \frac{2x}{1-x^2} \right)$. Here, $x = \frac{1}{3}$. Since $-1 < \frac{1}{3} < 1$, the formula is applicable.

    $2 \tan^{-1} \left( \frac{1}{3} \right) = \tan^{-1} \left( \frac{2 \left( \frac{1}{3} \right)}{1 - \left( \frac{1}{3} \right)^2} \right)$

    $= \tan^{-1} \left( \frac{\frac{2}{3}}{1 - \frac{1}{9}} \right)$

    $= \tan^{-1} \left( \frac{\frac{2}{3}}{\frac{9-1}{9}} \right)$

    $= \tan^{-1} \left( \frac{\frac{2}{3}}{\frac{8}{9}} \right)$

    To simplify the fraction, we multiply the numerator by the reciprocal of the denominator:
    

    $= \tan^{-1} \left( \frac{2}{3} \times \frac{9}{8} \right)$

    $= \tan^{-1} \left( \frac{18}{24} \right)$

    $= \tan^{-1} \left( \frac{3}{4} \right)$

    So, the original expression becomes $\tan^{-1} \left( \frac{3}{4} \right) + \tan^{-1} \left( \frac{1}{7} \right)$.
    

    2. Combine the two $\tan^{-1}$ terms:

    Now we have an expression of the form $\tan^{-1} x + \tan^{-1} y$, where $x = \frac{3}{4}$ and $y = \frac{1}{7}$. We use the formula $\tan^{-1} x + \tan^{-1} y = \tan^{-1} \left( \frac{x+y}{1-xy} \right)$.

    First, we check the condition $xy < 1$:

    $xy = \left( \frac{3}{4} \right) \left( \frac{1}{7} \right) = \frac{3}{28}$. Since $\frac{3}{28} < 1$, the formula is applicable. …

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