Question

If $\alpha +\beta =\frac{\pi }{2}$ and $\beta +\gamma =\alpha ,$ then find the value of tan $\alpha$.

Answer 1

$\alpha +\beta =\frac{\pi }{2}...........\left(1\right)\beta +\gamma =\alpha \Rightarrow \alpha -\beta =\gamma ...........\left(2\right)$

Adding (1) and (2)

$2\alpha =\frac{\pi }{2}+\gamma \Rightarrow \alpha =\frac{\pi }{4}+\frac{\gamma }{2}$

Now, $\tan \alpha =\tan (\frac{\pi }{4}+\frac{\gamma }{2})=\frac{\tan \frac{\pi }{4}+\tan \frac{\gamma }{2}}{1-\tan \frac{\pi }{4}\tan \frac{\gamma }{2}}=\frac{1+\tan \frac{\gamma }{2}}{1-\tan \frac{\gamma }{2}}$