The value of `sin^(2)acos ^(2)a( sec^(2) a+ cosec^(2) a )` is

Options

A. 2
B. 4
C. 1
D. 0

Answer

To solve the expression \( \sin^2 a \cos^2 a (\sec^2 a + \csc^2 a) \), we will break it down step by step.

### Step 1: Rewrite the expression
We start with the expression:
\[ \sin^2 a \cos^2 a (\sec^2 a + \csc^2 a) \]
We know that:
\[ \sec^2 a = \frac{1}{\cos^2 a} \quad \text{and} \quad \csc^2 a = \frac{1}{\sin^2 a} \]
So we can rewrite the expression as:
\[ \sin^2 a \cos^2 a \left( \frac{1}{\cos^2 a} + \frac{1}{\sin^2 a} \right) \]

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

### Step 3: Substitute back into the expression
Substituting this back into our expression gives:
\[ \sin^2 a \cos^2 a \cdot \frac{1}{\sin^2 a \cos^2 a} \]

### Step 4: Cancel out terms
Now we can cancel \( \sin^2 a \cos^2 a \):
\[ 1 \]

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