Question

If in a $\Delta ABC,\angle C=90{,}^{\circ }$ the find the maximum value of $\sin A\sin B$.

Answer 1

Let $y=\sin A.\sin B$
Now given $\angle C={90}^0=>A+B={90}^0$
$\Rightarrow y=\sin A.\sin ({90}^0-A)=\sin A.\cos A=\frac{1}{2}\sin 2A$
Hence maximum value of $y$ is $\frac{1}{2}$