In a ∆ABC, if
Question:

In a ∆ABC, if c2 sin A sin B = ab, then A + B = ______________.

Solution:

Since $c^{2} \sin A \sin B=a b$

$\Rightarrow c^{2}=\frac{a b}{\sin A \sin B}$

$=\frac{a}{\sin A} \times \frac{b}{\sin B}$

$c^{2}=\left(\frac{a}{\sin A}\right)^{2} \quad\left(\right.$ since by sine rule $\left.\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\right)$

$\Rightarrow c=\frac{a}{\sin A}=\frac{b}{\sin B}$

$\Rightarrow a=c \sin A$ and $b=c \sin B$

i. e $\sin A=\frac{a}{c}$ and $\sin B=\frac{b}{c}$

$\Rightarrow \Delta$ is a right angle triangle at $C$

$\Rightarrow A+B=\frac{\pi}{2}$