If $$2\sin \theta + 15 \cos^2 \theta = 7, 0^\circ < \theta < 90^\circ$$ then what is the value of $$\frac{3 - \tan \theta}{2 + \tan \theta}$$?

Given that  $$2 \sin \theta + 15 cos^2  \theta =7 $$

$$\Rightarrow 2\sin \theta  + 15 ( 1- \sin^2 \theta ) - 7 = 0 $$

$$\Rightarrow 15 \sin^2 \theta - 2 \sin \theta - 8 = 0 $$

$$ \Rightarrow 15 \sin^2 \theta + 10 \sin \theta - 12 \sin \theta - 8 = 0 $$ 

$$\Rightarrow  (3 \sin \theta + 2) (5\sin \theta - 4) = 0 $$ 

$$\Rightarrow \sin \theta = \dfrac{4}{5} $$ ,$$ \sin \theta = - \dfrac{2}{3} $$

then valid value $$\sin \theta = \dfrac {4}{5}$$ 

so  $$\cos \theta = \dfrac {3}{4} $$, $$\tan \theta = \dfrac {4}{3}$$ 

hence $$\dfrac {3 - \tan \theta} {2 + \tan \theta }$$

$$\Rightarrow \dfrac {3 - \dfrac{4}{3}} { 2 + \dfrac {4} {3}}$$ 

$$\Rightarrow \dfrac {9- 4}{6+4}$$

$$\Rightarrow \dfrac {5}{10} $$

$$\Rightarrow \dfrac {1}{2} $$ Ans