Question

If $\tan \left(\frac{\alpha r}{4}\right)=\cot \left(\frac{\beta r}{4}\right)$, then

• A. $\alpha +\beta =0$
• B. $\alpha +\beta =2n$
• C. $\alpha +\beta =2n+1$
• D. $\alpha +\beta =2(2n+1),\forall n$ is an integer

Answer 1

The correct option is D $\alpha +\beta =2(2n+1),\forall n$ is an integer

Since, $\tan \left(\frac{\alpha \pi }{4}\right)=\cot \left(\frac{\beta \pi }{4}\right)$
$\Rightarrow \tan \left(\frac{\alpha \pi }{4}\right)=\tan (\frac{\pi }{2}-\frac{\beta \pi }{4})$
$\Rightarrow \frac{\alpha \pi }{4}=n\pi +\frac{\pi }{2}-\frac{\beta \pi }{4}$
$\Rightarrow (\alpha +\beta )\frac{\pi }{4}=\left(d\frac{2n+1}{2}\right)\pi$
$\Rightarrow \alpha +\beta =2(2n+1),\forall n\in 1$