If `f(theta) = sin(tan^(-1)(sintheta/sqrt(cos2theta)))`, where `-pi/4 lt theta lt pi/4,` then the value of `d/(d(tantheta))\ f(theta)` is

Explanation

To find the value of \(\frac{d}{d(\tan \theta)} f(\theta)\) where \(f(\theta) = \sin(\tan^{-1}(\frac{\sin \theta}{\sqrt{\cos 2\theta}}))\), we will follow these steps:

### Step 1: Rewrite \(f(\theta)\)
Given:
\[ f(\theta) = \sin\left(\tan^{-1}\left(\frac{\sin \theta}{\sqrt{\cos 2\theta}}\right)\right) \]
We know that \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\). Thus, we can rewrite \(f(\theta)\) as:
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