`"If "f(theta)=tan (sin^(-1)sqrt((2)/(3+cos 2theta)))," then " fi nd f'(theta) and `

Options

A. `f((pi)/(4))=1`
B. `f((pi)/(4))=sqrt(2)`
C. `(d(f(theta)))/(d(cos theta))" at "theta=(pi)/(4)is -2`
D. `f'((pi)/(4))=sqrt(2)`

Explanation

To solve the problem, we need to find the derivative of the function \( f(\theta) = \tan(\sin^{-1}(\sqrt{\frac{2}{3 + \cos(2\theta)}})) \) and then evaluate \( \frac{d f}{d \cos(\theta)} \) at \( \theta = \frac{\pi}{4} \).

### Step 1: Rewrite the function using trigonometric identities

We start with the function:
\[ f(\theta) = \tan\left(\sin^{-1}\left(\sqrt{\frac{2}{3 + \cos(2\theta)}}\right)\right) \]
...