Question

Assertion :$co{s}^{-1}x=2si{n}^{-1}\sqrt{\frac{1-x}{2}}=2co{s}^{-1}\sqrt{\frac{1+x}{2}}$ Reason: $si{n}^{-1}(-x)=-si{n}^{-1}x,co{s}^{-1}(-x)=\pi -co{s}^{-1}x(-1\le x\le 1)$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Let $x=cosy$

$1-cosy=2si{n}^2\left(\frac{y}{2}\right)$

$y=2si{n}^{-1}\sqrt{\frac{1-cosy}{2}}$

Hence

$co{s}^{-1}\left(x\right)=2si{n}^{-1}(\sqrt{\frac{1-x}{2})}$

Similarly

$1+cosy=2co{s}^2\left(\frac{y}{2}\right)$

$y=2co{s}^{-1}\left(\sqrt{\frac{1+cosy}{2}}\right)$

$co{s}^{-1}\left(x\right)=2co{s}^{-1}\left(\sqrt{\frac{1+x}{2}}\right)$

$co{s}^{-1}\left(x\right)=2si{n}^{-1}(\sqrt{\frac{1-x}{2})}=2co{s}^{-1}(\sqrt{\frac{1+x}{2}})$

Hence Assertion is correct

Reason:

$si{n}^{-1}(-x)=-si{n}^{-1}x,co{s}^{-1}(-x)=\pi -co{s}^{-1}x(-1\le x\le 1)$

it is correct statement but not correct explanation of $A$

Hence the answer is $\left[B\right]$