If 0 leqslant cos ^{-1} x leqslant pi and fra | vedclass

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Question

(A) We are given the expression $\cos(2 \cos^{-1} x + \sin^{-1} x)$.We can rewrite this as $\cos(\cos^{-1} x + (\cos^{-1} x + \sin^{-1} x))$.Using the

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$-\sqrt{\frac{24}{25}}$

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(A) We are given the expression $\cos(2 \cos^{-1} x + \sin^{-1} x)$.We can rewrite this as $\cos(\cos^{-1} x + (\cos^{-1} x + \sin^{-1} x))$.Using the identity $\cos^{-1} x + \sin^{-1} x = \frac{\pi}{2}$ for $x \in [-1, 1]$,the expression becomes $\cos(\cos^{-1} x + \frac{\pi}{2})$.Using the trigonometric identity $\cos(	heta + \frac{\pi}{2}) = -\sin 	heta$,we get $-\sin(\cos^{-1} x)$.Since $\sin(\cos^{-1} x) = \sqrt{1 - x^2}$,the expression simplifies to $-\sqrt{1 - x^2}$.Given $x = \frac{1}{5}$,we substitute this value into the simplified expression:$-\sqrt{1 - (\frac{1}{5})^2} = -\sqrt{1 - \frac{1}{25}} = -\sqrt{\frac{24}{25}}$.Thus,the correct option is $A$.

If $0 \leqslant \cos ^{-1} x \leqslant \pi$ and $\frac{-\pi}{2} \leqslant \sin ^{-1} x \leqslant \frac{\pi}{2}$,then at $x=\frac{1}{5}$ the value of $\cos \left(2 \cos ^{-1} x+\sin ^{-1} x\right)$ is

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