If `A=29^@` , find the value of `sin^4 A+ cos^4 A + 2 sin^2 A cos^2 A`

Answer

To solve the problem, we need to find the value of the expression \( \sin^4 A + \cos^4 A + 2 \sin^2 A \cos^2 A \) where \( A = 29^\circ \).

### Step 1: Rewrite the expression
We can rewrite \( \sin^4 A \) and \( \cos^4 A \) in terms of squares:
\[ \sin^4 A = (\sin^2 A)^2 \quad \text{and} \quad \cos^4 A = (\cos^2 A)^2 \]
Thus, we can rewrite the expression as:
\[ \sin^4 A + \cos^4 A + 2 \sin^2 A \cos^2 A = (\sin^2 A)^2 + (\cos^2 A)^2 + 2 \sin^2 A \cos^2 A \]

### Step 2: Recognize the identity
Notice that this expression resembles the expansion of a square:
\[ a^2 + b^2 + 2ab = (a + b)^2 \]
Here, let \( a = \sin^2 A \) and \( b = \cos^2 A \). Therefore, we can rewrite the expression as:
\[ (\sin^2 A + \cos^2 A)^2 \]

### Step 3: Use the Pythagorean identity
From the Pythagorean identity, we know that:
\[ \sin^2 A + \cos^2 A = 1 \]
So, substituting this into our expression gives:
\[ (\sin^2 A + \cos^2 A)^2 = 1^2 = 1 \]

### Conclusion
Thus, the value of \( \sin^4 A + \cos^4 A + 2 \sin^2 A \cos^2 A \) is:
\[ \boxed{1} \]