Problem
Solve: \( \cos\left(2\sin^{-1}(-x)\right) = 0 \)
Solution
Use identity:
\[ \cos(2\theta) = 1 – 2\sin^2\theta \]
Let \( \theta = \sin^{-1}(-x) \)
Then:
\[ \sin \theta = -x \]
Now,
\[ \cos\left(2\sin^{-1}(-x)\right) = 1 – 2(-x)^2 \]
\[ = 1 – 2x^2 \]
Given:
\[ 1 – 2x^2 = 0 \]
\[ 2x^2 = 1 \]
\[ x^2 = \frac{1}{2} \]
\[ x = \pm \frac{1}{\sqrt{2}} \]
Final Answer
\[ \boxed{x = \pm \frac{1}{\sqrt{2}}} \]
Explanation
We used the identity cos(2θ) = 1 − 2sin²θ and simplified using sinθ = −x.