NIMCET 2022 — Mathematics PYQ

1. Definition and Range:

Let ${\cos }^{-1}(-x)=\theta$. By definition of inverse trigonometric functions, the range of the principal value of ${\cos }^{-1}x$ is $[0,\pi ]$. Therefore, $0\le \theta \le \pi$.

2. Rearranging the Equation:

From ${\cos }^{-1}(-x)=\theta$, we can write:

3. Using Trigonometric Identities:

Using the identity $\cos (\pi -\theta )=-\cos \theta$, we can substitute:

4. Taking the Inverse:

Now, take the inverse cosine on both sides:

5. Solving for $\theta$:

Rearrange the equation to isolate $\theta$:

Since we started with $\theta ={\cos }^{-1}(-x)$, we conclude:

Correct Option:

B. $\pi -{\cos }^{-1}x$