If cos θ=34, then find the value of 9tan2 θ + 9.

Given:

$\cos \theta=\frac{3}{4}$

$\Rightarrow \frac{1}{\cos \theta}=\frac{4}{3}$

$\Rightarrow \sec \theta=\frac{4}{3}$

We know that,

$\sec ^{2} \theta-\tan ^{2} \theta=1$

$\Rightarrow\left(\frac{4}{3}\right)^{2}-\tan ^{2} \theta=1$

$\Rightarrow \tan ^{2} \theta=\frac{16}{9}-1$

$\Rightarrow \tan ^{2} \theta=\frac{7}{9}$

Therefore,

$9 \tan ^{2} \theta+9=9 \times \frac{7}{9}+9$

$=7+9$

$=16$