# If sin α + sin β=a and cos α−cos β=b

Question:

If $\sin \alpha+\sin \beta=a$ and $\cos \alpha-\cos \beta=b$ then $\tan \frac{\alpha-\beta}{2}=$

(a) $-\frac{a}{b}$

(b) $-\frac{b}{a}$

(c) $\sqrt{a^{2}+b^{2}}$

(d) none of these

Solution:

(b) $-\frac{b}{a}$

Given:

$\sin \alpha+\sin \beta=a$

$\Rightarrow 2 \sin \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}=a \quad \ldots(1)$

Also,

$\cos \alpha+\cos \beta=b$

$\Rightarrow-2 \sin \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2}=b \quad \ldots(2)$

On dividing $(1)$ by $(2)$, we get

$\frac{-\cos \frac{\alpha-\beta}{2}}{\sin \frac{\alpha-\beta}{2}}=\frac{a}{b}$

$\Rightarrow \frac{-\sin \frac{\alpha-\beta}{2}}{\cos \frac{\alpha-\beta}{2}}=\frac{b}{a}$

$\Rightarrow \tan \frac{\alpha-\beta}{2}=-\frac{b}{a}$