If 4tan θ = 3 then (cos2 θ – sin2 θ) = ?

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,

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

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

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

$\Rightarrow H^{2}=25$

$\Rightarrow H=5$

Therefore,

$\sin \theta=\frac{P}{H}=\frac{3}{5}$

$\cos \theta=\frac{B}{H}=\frac{4}{5}$

$\cos ^{2} \theta-\sin ^{2} \theta=\left(\frac{4}{5}\right)^{2}-\left(\frac{3}{5}\right)^{2}$

$=\frac{16}{25}-\frac{9}{25}$

$=\frac{16-9}{25}$

$=\frac{7}{25}$

Hence, the correct option is (b).