What is the value of `cot^(2) theta-1/(sin^(2) theta)`?

Explanation

To solve the expression \( \frac{\cot^2 \theta - 1}{\sin^2 \theta} \), we will follow these steps:

### Step 1: Rewrite cotangent in terms of sine and cosine
We know that:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]
Thus,
\[ \cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} \]

### Step 2: Substitute cotangent in the expression
Substituting \( \cot^2 \theta \) into the expression gives:
\[ \frac{\cot^2 \theta - 1}{\sin^2 \theta} = \frac{\frac{\cos^2 \theta}{\sin^2 \theta} - 1}{\sin^2 \theta} \]

### Step 3: Combine the terms in the numerator
To combine the terms in the numerator, we need a common denominator:
\[ \frac{\cot^2 \theta - 1}{\sin^2 \theta} = \frac{\frac{\cos^2 \theta - \sin^2 \theta}{\sin^2 \theta}}{\sin^2 \theta} = \frac{\cos^2 \theta - \sin^2 \theta}{\sin^4 \theta} \]

### Step 4: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \cos^2 \theta = 1 - \sin^2 \theta \]
Substituting this into our expression gives:
\[ \cos^2 \theta - \sin^2 \theta = (1 - \sin^2 \theta) - \sin^2 \theta = 1 - 2\sin^2 \theta \]

### Step 5: Substitute back into the expression
Now substituting this back into our expression:
\[ \frac{1 - 2\sin^2 \theta}{\sin^4 \theta} \]

### Step 6: Final expression
Thus, the final value of the expression \( \frac{\cot^2 \theta - 1}{\sin^2 \theta} \) simplifies to:
\[ \frac{1 - 2\sin^2 \theta}{\sin^4 \theta} \]

### Conclusion
The value of \( \frac{\cot^2 \theta - 1}{\sin^2 \theta} \) is \( \frac{1 - 2\sin^2 \theta}{\sin^4 \theta} \).

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