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Question: If \(u = \log{\tan\left( \frac{\pi}{4} + \frac{x}{2} \right)}\),then the value of \({\tan h}\frac{u}...

If u=log⁡tan⁡(π4+x2)u = \log{\tan\left( \frac{\pi}{4} + \frac{x}{2} \right)},then the value of tan⁡hu2{\tan h}\frac{u}{2} is.

A

cot⁡x2\cot\frac{x}{2}

B

−cot⁡x2- \cot\frac{x}{2}

C

−tan⁡x2- \tan\frac{x}{2}

D

tan⁡x2\tan\frac{x}{2}

Answer

tan⁡x2\tan\frac{x}{2}

Explanation

Solution

u=log⁡tan⁡(π4+x2)=log⁡(1+tan⁡x21−tan⁡x2)u = \log{\tan\left( \frac{\pi}{4} + \frac{x}{2} \right)} = \log\left( \frac{1 + \tan\frac{x}{2}}{1 - \tan\frac{x}{2}} \right)

=2tanh⁡−1(tan⁡x2)= 2\tanh^{- 1}\left( \tan\frac{x}{2} \right) ⇒ tan⁡h (u2)=tan⁡x2{\tan h}\ \left( \frac{u}{2} \right) = \tan\frac{x}{2}.