Q.State True or False: The value of the expression $(\cos^{-1}x)^2$ is equal to $\sec^2 x$.

Concept: Inverse trigonometric functions — $\cos^{-1}x$ is the inverse cosine (output is an angle), while $\sec x$ is the reciprocal of $\cos x$ (output is a ratio). They are fundamentally different kinds of functions.

Step 1: Let $y = \cos^{-1}x$. Then $x = \cos y$, and $y \in [0, \pi]$. The expression $(\cos^{-1}x)^2$ is the square of an angle.

Step 2: The expression $\sec^2 x$ means $(\sec x)^2 = \frac{1}{\cos^2 x}$, which is a function of the variable $x$ (treated as an angle here). This is a ratio, not an angle squared. …

Concept and Intuition

The core confusion here is between inverse trigonometric functions and reciprocal trigonometric functions. Many students mix up $\cos^{-1}x$ (which means "the angle whose cosine is $x$") with $\sec x$ (which is $\frac{1}{\cos x}$). The notation itself is partly to blame: the $-1$ superscript in $\cos^{-1}x$ looks like an exponent, but it actually denotes the inverse function, not the reciprocal.

Let’s be crystal clear:

• $\cos^{-1}x$ is the inverse cosine (also written $\arccos x$). It takes a number $x$ (where $-1 \le x \le 1$) and returns an angle $\theta$ such that $\cos\theta = x$ and $0 \le \theta \le \pi$.
• $\sec x$ is the secant of $x$, defined as $\frac{1}{\cos x}$. It takes an angle $x$ and returns a real number (provided $\cos x \neq 0$).

So $(\cos^{-1}x)^2$ is the square of an angle, while $\sec^2 x$ is the square of a ratio. They live in different worlds.

Step-by-Step Reasoning

1. Understand the domains.

• $(\cos^{-1}x)^2$ is defined only when $x \in [-1, 1]$, because $\cos^{-1}x$ is defined only for those $x$.
• $\sec^2 x$ is defined for all real $x$ except where $\cos x = 0$, i.e., $x \neq \frac{\pi}{2} + n\pi$, $n \in \mathbb{Z}$.

These domains are completely different. For example, take $x = 0.5$:

• $(\cos^{-1}0.5)^2 = (\frac{\pi}{3})^2 = \frac{\pi^2}{9} \approx 1.0966$
• $\sec^2(0.5) = \frac{1}{\cos^2(0.5)} \approx \frac{1}{0.8776^2} \approx 1.298$

They are not equal.

2. Check a specific value to see the absurdity.

Take $x = 1$:

• $(\cos^{-1}1)^2 = (0)^2 = 0$
• $\sec^2(1) = \frac{1}{\cos^2(1)} \approx \frac{1}{0.5403^2} \approx 3.425$

Clearly $0 \neq 3.425$. So the statement is false.

3. Understand the deeper reason: inverse vs. reciprocal. …

Method: Distinguish an inverse function from a reciprocal (power)

Steps

Step 1: Read the notation precisely.

$\cos^{-1}x$ means the inverse cosine — an angle in $[0,\pi]$ — NOT $(\cos x)^{-1}=\sec x$.

Step 2: Compare the two objects. …

Common Mistakes

Mistake 1: Reading $\cos^{-1}x$ as $\frac{1}{\cos x}=\sec x$.

Why it's wrong: the $-1$ denotes the inverse function, not the reciprocal. Correct approach: the reciprocal $\frac{1}{\cos x}$ is written $\sec x$ or $(\cos x)^{-1}$, a different object from $\cos^{-1}x$. …