The value of `sin^2 theta cos^2 theta (sec^2 theta + cosec^2 theta )` is

Options

A. 2
B. 4
C. 1
D. 3

Answer

To find the value of the expression \( \sin^2 \theta \cos^2 \theta (\sec^2 \theta + \csc^2 \theta) \), we can follow these steps:

### Step 1: Rewrite the expression
The expression can be rewritten as:
\[ \sin^2 \theta \cos^2 \theta (\sec^2 \theta + \csc^2 \theta) \]

### Step 2: Substitute the definitions of secant and cosecant
Recall that:
\[ \sec^2 \theta = \frac{1}{\cos^2 \theta} \quad \text{and} \quad \csc^2 \theta = \frac{1}{\sin^2 \theta} \]
Substituting these into the expression gives:
\[ \sin^2 \theta \cos^2 \theta \left(\frac{1}{\cos^2 \theta} + \frac{1}{\sin^2 \theta}\right) \]

### Step 3: Simplify the expression inside the parentheses
Now, we can simplify the expression inside the parentheses:
\[ \frac{1}{\cos^2 \theta} + \frac{1}{\sin^2 \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin^2 \theta \cos^2 \theta} \]
Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we have:
\[ \frac{1}{\sin^2 \theta \cos^2 \theta} \]

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

### Step 5: Simplify the expression
The terms \( \sin^2 \theta \cos^2 \theta \) cancel out:
\[ = 1 \]

### Final Answer
Thus, the value of the expression \( \sin^2 \theta \cos^2 \theta (\sec^2 \theta + \csc^2 \theta) \) is:
\[ \boxed{1} \]

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