Question

If $2\sec 2\theta =\tan \varphi +\cot \varphi$, then one of the values of $\theta +\varphi$ is

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. none of these

Answer 1

Answer: (B)

$\frac{\pi }{4}$

$2\sec 2\theta =\tan \varphi +cot\varphi$

$\Rightarrow 2\sec 2\theta =\frac{{\sin }^2\varphi +{\cos }^2\varphi }{\sin \varphi \cos \varphi }$

$\Rightarrow 2\sec 2\theta =\frac{1}{\sin \varphi \cos \varphi }$

$\Rightarrow 2\sin \varphi \cos \varphi =\cos 2\theta$

$\Rightarrow \sin 2\varphi =\cos 2\theta$

$\Rightarrow \sin 2\varphi =\sin (\frac{\pi }{2}-2\theta )$

$\therefore 2\theta +2\varphi =\frac{\pi }{2}$
$\therefore \theta +\varphi =\frac{\pi }{4}$