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

$\frac{1}{2}$

2 sinθ + 15 cos²θ = 7

{ using , sin²θ  + cos²θ  = 1 }

2 sinθ + 15( 1 - sin²θ ) = 7

15 sin²θ - 2sinθ - 8 = 0

15 sin²θ - 12sinθ + 10sinθ  - 8 = 0

3 sinθ (5 sinθ - 4 ) + 2 (5 sinθ - 4 )= 0

Either (3 sinθ + 2 )= 0  OR  (5 sinθ - 4 )= 0 

sinθ = - \(\frac{2}{3}\)   { not possible }

So , sinθ =  \(\frac{4}{5}\)

By using pythagoras theorem ,

P² + B² = H²

4² + B² = 5²

B = 3

Now,

\(\frac{3 - tanθ}{2 + tanθ}\)

= \(\frac{3 - 4/3}{2 +4/3}\)

= \(\frac{1}{2}\)