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Question: If \(\sin^{- 1}x + \cot^{- 1}\left( \frac{1}{2} \right) = \frac{\pi}{2},\) then x is...

If sin⁡−1x+cot⁡−1(12)=π2,\sin^{- 1}x + \cot^{- 1}\left( \frac{1}{2} \right) = \frac{\pi}{2}, then x is

A

0

B

15\frac{1}{\sqrt{5}}

C

25\frac{2}{\sqrt{5}}

D

32\frac{\sqrt{3}}{2}

Answer

15\frac{1}{\sqrt{5}}

Explanation

Solution

sin⁡−1x+cot⁡−1(x2)=π2\sin^{- 1}x + \cot^{- 1}\left( \frac{x}{2} \right) = \frac{\pi}{2} (∵cot⁡−112=cos⁡−115)\left( \because\cot^{- 1}\frac{1}{2} = \cos^{- 1}\frac{1}{\sqrt{5}} \right)

sin⁡−1x+cos⁡−115=π2\sin^{- 1}x + \cos^{- 1}\frac{1}{\sqrt{5}} = \frac{\pi}{2}; Clearly, x=15x = \frac{1}{\sqrt{5}}