Question

If $\alpha$, $\beta$ and $\gamma$ are in A.P., $\frac{\sin \alpha -\sin \gamma }{\cos \gamma -\cos \alpha }$ equals to

• A. $\tan \beta$
• B. $\cot \beta$
• C. $\tan \frac{\beta }{2}$
• D. $\cot \frac{\beta }{2}$

Answer 1

The correct option is B

$\cot \beta$

$\frac{\sin \alpha -\sin \gamma }{\cos \gamma -\cos \alpha }$ = $\frac{2\sin \frac{\alpha -\gamma }{2}\cos \frac{\alpha +\gamma }{2}}{2\sin \frac{\alpha -\gamma }{2}\sin \frac{\alpha +\gamma }{2}}$

= $\cot \left(\frac{\alpha +\gamma }{2}\right)$
$\text{Since}\alpha$, $\beta$ and $\gamma$ are in A.P., so $\beta =\frac{\alpha +\gamma }{2}$
Thus, substituting back we get
$\frac{\sin \alpha -\sin \gamma }{\cos \gamma -\cos \alpha }=\cot \beta$.