Find the value of sec²(cot⁻¹ 1/3) + cosec²(tan⁻¹ 1/2).

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Q. Find the value of sec²(cot⁻¹ 1/3) + cosec²(tan⁻¹ 1/2).

West Bengal WbchseWest Bengal HS (WBCHSE) Board 2022Subjective· 2mImportance★★★★★

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Concept understanding — Inverse Trigonometric Functions

Why an inverse needs a restricted domain. A function has an inverse only where it is one-to-one and onto. None of the six trigonometric functions is one-to-one on its natural domain, because each is periodic — infinitely many angles give the same sine, the same cosine, and so on. (Compare $y=x^2$ on all of $\mathbb R$ : it isn't one-to-one either, since $x=2$ and $x=-2$ both give $y=4$ ; only once the domain is cut down to $x\ge0$ does an inverse, $\sqrt x$ , exist.) So before " $\sin^{-1}$ ", " $\cos^{-1}$ ", etc. can mean anything, each trigonometric function is first restricted to one conventional interval on which it is one-to-one and whose image is its whole natural range. That restricted interval always contains $0$ together with some positive angles.

The six function/inverse domain-range pairs.

Function (restricted)	Domain $\to$ Range	Inverse	Domain $\to$ Range
$\sin x$	$\left[-\tfrac\pi2,\tfrac\pi2\right]\to[-1,1]$	$\sin^{-1}x$	$[-1,1]\to\left[-\tfrac\pi2,\tfrac\pi2\right]$
$\cos x$	$[0,\pi]\to[-1,1]$	$\cos^{-1}x$	$[-1,1]\to[0,\pi]$
$\tan x$	$\left(-\tfrac\pi2,\tfrac\pi2\right)\to(-\infty,\infty)$	$\tan^{-1}x$	$(-\infty,\infty)\to\left(-\tfrac\pi2,\tfrac\pi2\right)$
$\cot x$	$(0,\pi)\to(-\infty,\infty)$	$\cot^{-1}x$	$(-\infty,\infty)\to(0,\pi)$
$\csc x$	$\left[-\tfrac\pi2,\tfrac\pi2\right]-\{0\}\to\mathbb R-(-1,1)$	$\csc^{-1}x$	$\mathbb R-(-1,1)\to\left[-\tfrac\pi2,\tfrac\pi2\right]-\{0\}$
$\sec x$	$[0,\pi]-\left\{\tfrac\pi2\right\}\to\mathbb R-(-1,1)$	$\sec^{-1}x$	$\mathbb R-(-1,1)\to[0,\pi]-\left\{\tfrac\pi2\right\}$

Each inverse function's stated domain is exactly the range of the original restricted function, and its range is exactly that restricted domain — an inverse always swaps domain and range with the function it undoes, and its graph is the mirror image of the original's graph in the line $y=x$ . …

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Convert each inverse-trig value to a right-triangle ratio and use $\sec^2\theta = 1+\tan^2\theta$ and $\csc^2\phi = 1+\cot^2\phi$ .

Term 1: $\sec^2(\cot^{-1}\frac13)$ .

Let $\theta = \cot^{-1}\frac13$ , so $\cot\theta = \frac13$ , hence $\tan\theta = 3$ .

Using the identity $\sec^2\theta = 1+\tan^2\theta$ :

$\sec^2\theta = 1 + 3^2 = 1+9 = 10$

Term 2: $\csc^2(\tan^{-1}\frac12)$ . …

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Q.Find the value of sec²(cot⁻¹ 1/3) + cosec²(tan⁻¹ 1/2).

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