Question:
If $4 \tan \theta=3$ then prove that $\sin \theta \cos \theta=\frac{12}{25}$.
Solution:
Given: $4 \tan \theta=3$
$\Rightarrow \tan \theta=\frac{3}{4}$
Since, $\tan \theta=\frac{P}{B}$
$\Rightarrow P=3$ and $B=4$
Using Pythagoras theorem,
$P^{2}+B^{2}=H^{2}$
$\Rightarrow 3^{2}+4^{2}=H^{2}$
$\Rightarrow H^{2}=9+16=25$
$\Rightarrow H=5$
Therefore,
$\sin \theta=\frac{P}{H}=\frac{3}{5}$
$\cos \theta=\frac{B}{H}=\frac{4}{5}$
$\sin \theta \times \cos \theta=\frac{3}{5} \times \frac{4}{5}$
$=\frac{12}{25}$
Hence, $\sin \theta \times \cos \theta=\frac{12}{25}$.