If `1/2sin^(-1)[(3sin2theta)/(5+4cos2theta)]=tan^(-1)x ,t h e nx=` (a)`tan3theta` (b) `3tantheta` (c) `(1/3)tantheta` (d) `3cottheta`

Explanation

To solve the equation \( \frac{1}{2} \sin^{-1} \left( \frac{3 \sin 2\theta}{5 + 4 \cos 2\theta} \right) = \tan^{-1} x \), we will follow these steps:

### Step 1: Use the double angle identities
Recall the double angle identities:
- \( \sin 2\theta = 2 \sin \theta \cos \theta \)
- \( \cos 2\theta = 1 - 2 \sin^2 \theta \) or \( \cos 2\theta = 2 \cos^2 \theta - 1 \)

Using the identity for \( \sin 2\theta \) and \( \cos 2\theta \), we can express the equation in terms of \( \tan \theta \).
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