If $3 \tan \theta=2 \sqrt{3} \sin \theta, 0^{\circ}<\theta<90^{\circ}$, then find the value of $2 \sin ^2 2 \theta-3 \cos ^2 3 \theta$.
Answer & explanation
Correct answer: option 2
3 tanθ = 2√3 sinθ
cosθ = \(\frac{√3}{2}\)
{ we know, cos30º = \(\frac{√3}{2}\) }
Now,
2sin²2θ - 3cos²3θ
= 2sin²60º - 3cos²90º
= 2 × \(\frac{3}{4}\) - 0
= \(\frac{3}{2}\)