Solve `cosec^(2)theta-cot^(2) theta=cos theta`.

Explanation

To solve the equation \( \csc^2 \theta - \cot^2 \theta = \cos \theta \), we can start by using the Pythagorean identity for cosecant and cotangent.

### Step 1: Use the Pythagorean identity
Recall that:
\[ \csc^2 \theta - \cot^2 \theta = 1 \]
Thus, we can rewrite the equation:
\[ 1 = \cos \theta \]

### Step 2: Solve for \( \theta \)
The equation \( \cos \theta = 1 \) has solutions where:
\[ \theta = 2n\pi \quad \text{for } n \in \mathbb{Z} \]
This means that \( \theta \) can take values like \( 0, 2\pi, 4\pi, \ldots \)

### Step 3: Check for additional solutions
Now, we need to check if there are any other solutions. We can also analyze the original equation:
\[ \csc^2 \theta - \cot^2 \theta = \cos \theta \]
Substituting the identities:
\[ 1 = \cos \theta \]
This confirms that the only solutions are indeed from \( \cos \theta = 1 \).

### Conclusion
Thus, the final solution is:
\[ \theta = 2n\pi \quad \text{for } n \in \mathbb{Z} \]