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Question: The value of \({\cos h}^{- 1}(\sec x)\) is...

The value of cos⁡h−1(sec⁡x){\cos h}^{- 1}(\sec x) is

A

log⁡(1+sin⁡xcos⁡x)\log\left( \frac{1 + \sin x}{\cos x} \right)

B

log⁡(1−sin⁡xcos⁡x)\log\left( \frac{1 - \sin x}{\cos x} \right)

C

log⁡(1+cos⁡xsin⁡x)\log\left( \frac{1 + \cos x}{\sin x} \right)

D

log⁡(1−cos⁡xsin⁡x)\log\left( \frac{1 - \cos x}{\sin x} \right)

Answer

log⁡(1+sin⁡xcos⁡x)\log\left( \frac{1 + \sin x}{\cos x} \right)

Explanation

Solution

Here cos⁡h−1(sec⁡x)=log⁡(sec⁡x+sec⁡2x−1){\cos h}^{- 1}(\sec x) = \log(\sec x + \sqrt{\sec^{2}x - 1}) =

log⁡(sec⁡x+tan⁡x)\log(\sec x + \tan x) =log⁡(1+sin⁡xcos⁡x)\log\left( \frac{1 + \sin x}{\cos x} \right)