Q. 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) $\alpha + \beta = 2(2n + 1),\ \forall n$ is an integer
Explanation
(d)
Given, $\tan{\left( \frac{\alpha\pi}{4} \right) = \tan\left( \frac{\pi}{2} - \frac{\beta\pi}{4} \right)}$
$\Rightarrow \ \ \frac{\alpha\pi}{4} = n\pi + \frac{\pi}{2} - \frac{\beta\pi}{4}$
$\Rightarrow \ \ (\alpha + \beta)\frac{\pi}{4} = \left( \frac{2n + 1}{2} \right)\pi$
$\Rightarrow \ \ \ \ \alpha + \beta = 2(2n + 1),\ \forall\ n \in I$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.