Question

If ${}^{\prime }{x}^{\prime }$ is an acute angle and $y={\log }_e\left(\tan (\frac{\pi }{4}+\frac{x}{2})\right)$ then $\cos x.\cos y=$...........

• A. $0$
• B. $-1$
• C. $1$
• D. $2$

Answer 1

Answer: (C)

$1$

Given the question.....

$\begin{array}{c}y=lo{g}_e\left[tan(\frac{\pi }{4}+\frac{x}{2})\right]|\begin{array}{c}{\log }_{e}a=b\\ a=e^b\end{array}\\ \tan \left(\frac{\pi }{4}+\frac{x}{2}\right)=e^y\\ \cos x.\cos y=\cos x\frac{e^y+e^{-y}}{2}\\ =\cos x\frac{e^{-y}(e^{2y}+1)}{2}\\ =\cos x\left(\frac{e^{2y}+1}{2e^y}\right)|\sin 2\theta =\frac{2\tan \theta }{1+{\tan }^2\theta }\\ =\cos x.\frac{{\tan }^2\left(\frac{\pi }{4}+\frac{x}{2}\right)+1}{2\tan \left(\frac{\pi }{4}+\frac{x}{2}\right)}\\ =\cos x.\frac{1}{\sin 2\left(\frac{\pi }{4}+\frac{x}{2}\right)}=\cos x.\frac{1}{\cos x}=1\\ so,thevalueof\cos x.\cos y=1\\ andthecorrectoptionisC.\end{array}$