If + =a and - =b then - 2 = - Vidyadip

MCQ

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

A
$-\frac{\text{a}}{\text{b}}$
✓
$-\frac{\text{b}}{\text{a}}$
C
$\sqrt{\text{a}^2+\text{b}^2}$
D
None of these

✓

Answer

Correct option: B.

$-\frac{\text{b}}{\text{a}}$

Given:
$\sin\alpha+\sin\beta=\text{a}$
$\Rightarrow2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha+\beta}{2}=\text{a}\ .....(1)$
Also,
$\cos\alpha+\cos\beta=\text{b}$
$\Rightarrow-2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha+\beta}{2}=\text{b}\ .....(2)$
On dividing $(1)$ by $(2),$ we get
$\frac{-\cos\frac{\alpha+\beta}{2}}{\sin\frac{\alpha-\beta}{2}}=\frac{\text{a}}{\text{b}}$
$\Rightarrow\frac{-\sin\frac{\alpha+\beta}{2}}{\cos\frac{\alpha-\beta}{2}}=\frac{\text{b}}{\text{a}}$
$\Rightarrow\tan\frac{\alpha-\beta}{2}=-\frac{\text{b}}{\text{a}}$

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