`tan^(-1)(tansqrt(1-theta))=sqrt(1-theta)` when

Options

A. `-(pi)/(2) lt theta (pi)/(2)`
B. `theta gt (4-pi)/(4)`
C. `thetalt (4-pi)/(4)`
D. `(4-pi^(2))/(4) lt le 1`

Explanation

To solve the equation \( \tan^{-1}(\tan(\sqrt{1 - \theta})) = \sqrt{1 - \theta} \), we need to analyze the conditions under which this equality holds.

### Step 1: Understand the Range of the Functions
The function \( \tan^{-1}(x) \) gives an angle whose tangent is \( x \). The output of \( \tan^{-1}(x) \) is restricted to the range \( (-\frac{\pi}{2}, \frac{\pi}{2}) \).

### Step 2: Analyze the Input to the Tangent Function
The expression \( \sqrt{1 - \theta} \) must also lie within the range of the tangent function. Therefore, we need:
\[
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