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Question: If \(x = \frac{1}{5}\), the value of \(\cos\left( \cos^{- 1}x + 2\sin^{- 1}x \right)\) is...

If x=15x = \frac{1}{5}, the value of cos⁡(cos⁡−1x+2sin⁡−1x)\cos\left( \cos^{- 1}x + 2\sin^{- 1}x \right) is

A

−2425- \sqrt{\frac{24}{25}}

B

2425\sqrt{\frac{24}{25}}

C

−15- \frac{1}{5}

D

15\frac{1}{5}

Answer

15\frac{1}{5}

Explanation

Solution

The given expression is equal to

cos⁡(cos⁡−1x+sin⁡−1x)=cos⁡(π2+sin⁡−1x)\cos\left( \cos^{- 1}x + \sin^{- 1}x \right) = \cos\left( \frac{\pi}{2} + \sin^{- 1}x \right)

= −sin⁡(sin⁡−1x)=−x=−15- \sin\left( \sin^{- 1}x \right) = - x = - \frac{1}{5}