Solve the Following Questions

Question:

If $\left(\sin ^{-1} x\right)^{2}-\left(\cos ^{-1} x\right)^{2}=a ; 0

  1. $\cos \left(\frac{4 a}{\pi}\right)$

  2. $\sin \left(\frac{2 a}{\pi}\right)$

  3. $\cos \left(\frac{2 a}{\pi}\right)$

  4. $\sin \left(\frac{4 a}{\pi}\right)$


Correct Option: , 2

Solution:

Given $a=\left(\sin ^{-1} x\right)^{2}-\left(\cos ^{-1} x\right)^{2}$

$=\left(\sin ^{-1} x+\cos ^{-1} x\right)\left(\sin ^{-1} x-\cos ^{-1} x\right)$

$=\frac{\pi}{2}\left(\frac{\pi}{2}-2 \cos ^{-1} x\right)$

$\Rightarrow 2 \cos ^{-1} x=\frac{\pi}{2}-\frac{2 a}{\pi}$

$\Rightarrow \cos ^{-1}\left(2 x^{2}-1\right)=\frac{\pi}{2}-\frac{2 a}{\pi}$

$\Rightarrow 2 \mathrm{x}^{2}-1=\cos \left(\frac{\pi}{2}-\frac{2 \mathrm{a}}{\pi}\right)$ option (2)

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