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Question: If \(- \frac{\pi}{2} < x < \frac{\pi}{2}\), then the value of \({logsec}x\) is...

If −π2<x<π2- \frac{\pi}{2} < x < \frac{\pi}{2}, then the value of logsecx{logsec}x is

A

2coth⁡−1(cosec2x2−1)2\coth^{- 1}\left( \text{cose}\text{c}^{2}\frac{x}{2} - 1 \right)

B

2cot⁡h−1(cosec2x2+1)2{\cot h}^{- 1}\left( c\text{ose}\text{c}^{2}\frac{x}{2} + 1 \right)

C

2cosech-1(cot⁡2x2−1)2\text{cosec}\text{h}^{\text{-1}}\left( \cot^{2}\frac{x}{2} - 1 \right)

D

2cosech-1(cot⁡2x2+1)2\text{cosec}\text{h}^{\text{-1}}\left( \cot^{2}\frac{x}{2} + 1 \right)

Answer

2coth⁡−1(cosec2x2−1)2\coth^{- 1}\left( \text{cose}\text{c}^{2}\frac{x}{2} - 1 \right)

Explanation

Solution

Let logsecx=y{logsec}x = y; ∴1cos⁡x=ey/2e−y/2\therefore\frac{1}{\cos x} = \frac{e^{y/2}}{e^{- y/2}}

By componendo and Dividendo rule, 1+cos⁡x1−cos⁡x=ey/2+e−y/2ey/2−e−y/2\frac{1 + \cos x}{1 - \cos x} = \frac{e^{y/2} + e^{- y/2}}{e^{y/2} - e^{- y/2}}⇒ cot⁡2(x2)=cot⁡h(y2)\cot^{2}\left( \frac{x}{2} \right) = {\cot h}\left( \frac{y}{2} \right)

⇒ y=2cot⁡h−1(cosec2x2−1)y = 2{\cot h}^{- 1}\left( \text{cose}\text{c}^{2}\frac{x}{2} - 1 \right)