Prove that
Question:

If $\tan \theta=\frac{4}{3}$ then $(\sin \theta+\cos \theta)=?$

(a) $\frac{7}{3}$

(b) $\frac{7}{4}$

(c) $\frac{7}{5}$

(d) $\frac{5}{7}$

Solution:

Given : $\tan \theta=\frac{4}{3}$

Since, $\tan \theta=\frac{P}{B}$

$\Rightarrow P=4$ and $B=3$

Using Pythagoras theorem,

$P^{2}+B^{2}=H^{2}$

$\Rightarrow 4^{2}+3^{2}=H^{2}$

$\Rightarrow H^{2}=16+9$

$\Rightarrow H^{2}=25$

$\Rightarrow H=5$

Therefore,

$\sin \theta=\frac{P}{H}=\frac{4}{5}$

$\cos \theta=\frac{B}{H}=\frac{3}{5}$

Now,

$\sin \theta+\cos \theta=\frac{4}{5}+\frac{3}{5}$

$=\frac{7}{5}$

Hence, the correct option is (c).