Prove that `sec^(2) theta + cosec^(2) theta = sec^(2) theta * cosec^(2) theta`

Answer

To prove that \( \sec^2 \theta + \csc^2 \theta = \sec^2 \theta \cdot \csc^2 \theta \), we will start with the left-hand side (LHS) and manipulate it to show that it equals the right-hand side (RHS).

### Step-by-Step Solution:

1. **Start with the LHS**:
\[ \text{LHS} = \sec^2 \theta + \csc^2 \theta \]

2. **Rewrite secant and cosecant in terms of sine and cosine**:
\[ \sec^2 \theta = \frac{1}{\cos^2 \theta} \quad \text{and} \quad \csc^2 \theta = \frac{1}{\sin^2 \theta} \]
Thus, we can rewrite LHS as:
\[ \text{LHS} = \frac{1}{\cos^2 \theta} + \frac{1}{\sin^2 \theta} \]

3. **Find a common denominator**:
The common denominator for the fractions is \( \cos^2 \theta \sin^2 \theta \):
\[ \text{LHS} = \frac{\sin^2 \theta + \cos^2 \theta}{\cos^2 \theta \sin^2 \theta} \]

4. **Use the Pythagorean identity**:
We know from trigonometric identities that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Therefore, we can substitute this into our equation:
\[ \text{LHS} = \frac{1}{\cos^2 \theta \sin^2 \theta} \]

5. **Rewrite the expression**:
Now, we can rewrite this as:
\[ \text{LHS} = \frac{1}{\cos^2 \theta} \cdot \frac{1}{\sin^2 \theta} = \sec^2 \theta \cdot \csc^2 \theta \]

6. **Conclude the proof**:
Thus, we have shown that:
\[ \text{LHS} = \sec^2 \theta \cdot \csc^2 \theta \]
Therefore, we conclude that:
\[ \sec^2 \theta + \csc^2 \theta = \sec^2 \theta \cdot \csc^2 \theta \]