For real values of `theta,` which of the following is/are always positive? `"cos"(costheta)` (b) `cos(sintheta)` `sin(costheta)` (d) `sin(sintheta)`

Options

A. `cos(costheta)`
B. `cos(sintheta)`
C. `sin(costheta)`
D. `sin(sintheta)`

Explanation

To determine which of the given functions is/are always positive for real values of \(\theta\), we will analyze each function one by one.

### Step 1: Analyze \( \cos(\cos(\theta)) \)

1. The function \(\cos(\theta)\) oscillates between -1 and 1 for all real values of \(\theta\).
2. Therefore, the input to the outer cosine function, \(\cos(\theta)\), will always be in the range \([-1, 1]\).
3. The cosine function, \(\cos(x)\), is positive for \(x\) in the range \([-1, 1]\) since \(\cos(0) = 1\) and \(\cos(x)\) decreases from 1 to -1 as \(x\) goes from 0 to \(\pi\).
4. Hence, \(\cos(\cos(\theta))\) will always be positive.
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