Question

If $2\sec 2\alpha =\tan \beta +\cot \beta$, then one of the value of $\alpha +\beta$ is-

• A. $\frac{\pi }{4}$
• B. $\frac{\pi }{2}$
• C. $\pi$
• D. $n\pi -\frac{\pi }{4},nϵI$

Answer 1

The correct option is A $\frac{\pi }{4}$

$2\sec 2\alpha =\tan \beta +\cot \beta$

$\Rightarrow 2\sec 2\alpha =\frac{{\sec }^2\beta }{\tan \beta }$

$\Rightarrow 2\sec 2\alpha =\frac{{\sec }^2\beta \times \cos \beta }{\sin \beta }$

$\Rightarrow \sin 2\beta =\cos 2\alpha$

So, one of the values is $2\alpha =2\beta =\frac{\pi }{4}$

One of the solutions is $2\alpha +2\beta =\frac{\pi }{2}\Rightarrow \alpha +\beta =\frac{\pi }{4}$