Question

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

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

Answer 1

Answer: (D)

Since, $\tan \left(\frac{\alpha\pi}{4}\right) = \cot \left(\frac{\beta\pi}{4}\right) $

$\Rightarrow \tan \left(\frac{\alpha\pi}{4}\right) = \tan \left(\frac{\pi}{2} -\frac{\beta\pi}{4}\right) $

$\Rightarrow \frac{\alpha\pi}{4} = n\pi +\frac{\pi}{2} -\frac{\beta\pi}{4} $

$\Rightarrow \left(\alpha+\beta\right) \frac{\pi}{4} = \left(\frac{2n+1}{2}\right)\pi $

$ \Rightarrow \alpha +\beta = 2\left(2n+1\right), \forall n \in I$