Question

If $\alpha +\beta =\frac{\pi }{2}$ and $\beta +\gamma =\alpha ,$ then tan $\alpha$ equal to

• A. $2(\tan \beta +\tan \gamma )$
• B. $\tan \beta +\tan \gamma$
• C. $\tan \beta +2\tan \gamma$
• D. $2\tan \beta +\tan \gamma$

Answer 1

The correct option is C $\tan \beta +2\tan \gamma$

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

Again, $\beta +\gamma =\alpha$
$\Rightarrow \gamma =\alpha -\beta$
$\Rightarrow \tan \gamma =\frac{\tan \alpha -\tan \beta }{1+\tan \alpha \tan \beta }$
$\Rightarrow \tan \gamma =\frac{\tan \alpha -\tan \beta }{2}\left[\text{from eq.}\right(1\left)\right]$
$\Rightarrow \tan \alpha =\tan \beta +2\tan \gamma$