# If tan α=

Question:

If $\tan \alpha=\frac{1-\cos \beta}{\sin \beta}$, then

(a) $\tan 3 \alpha=\tan 2 \beta$

(b) $\tan 2 \alpha=\tan \beta$

(c) $\tan 2 \beta=\tan \alpha$

(d) none of these

Solution:

(b) $\tan 2 \alpha=\tan \beta$

$\tan \alpha=\frac{1-\cos \beta}{\sin \beta}$

$=\frac{2 \sin ^{2} \frac{\beta}{2}}{2 \sin \frac{\beta}{2} \cos \frac{\beta}{2}}$

$=\frac{\sin \frac{\beta}{2}}{\cos \frac{\beta}{2}}$

$\Rightarrow \tan \alpha=\tan \frac{\beta}{2}$

$\Rightarrow \alpha=\frac{\beta}{2}$

$\Rightarrow 2 \alpha=\beta$

$\therefore \tan 2 \alpha=\tan \beta$