`1+ cos ^(2) 2A` is equal to

Options

A. `sin ^(4) A+ cos ^(4) A`
B. `sin ^(2) 2A`
C. `2(cos ^(4) A + sin ^(4) A)`
D. `2 (cos ^(4) A- sin ^(4) A)`

Answer

To solve the expression \(1 + \cos^2(2A)\), we can start by using some trigonometric identities.

### Step 1: Use the Pythagorean Identity
We know from the Pythagorean identity that:
\[ \sin^2(x) + \cos^2(x) = 1 \]
This implies:
\[ \sin^2(2A) = 1 - \cos^2(2A) \]

### Step 2: Substitute into the Expression
We can rewrite the expression \(1 + \cos^2(2A)\) using the identity from Step 1:
\[ 1 + \cos^2(2A) = 1 + (1 - \sin^2(2A)) = 2 - \sin^2(2A) \]

### Step 3: Express \(\sin^2(2A)\) in terms of \(\sin^2(A)\) and \(\cos^2(A)\)
Using the double angle formula for sine:
\[ \sin(2A) = 2\sin(A)\cos(A) \]
Thus,
\[ \sin^2(2A) = (2\sin(A)\cos(A))^2 = 4\sin^2(A)\cos^2(A) \]

### Step 4: Substitute \(\sin^2(2A)\) back into the expression
Now we can substitute \(\sin^2(2A)\) into our expression:
\[ 1 + \cos^2(2A) = 2 - 4\sin^2(A)\cos^2(A) \]

### Step 5: Simplify the Expression
We can further simplify \(4\sin^2(A)\cos^2(A)\) using the identity:
\[ \sin^2(A)\cos^2(A) = \frac{1}{4}\sin^2(2A) \]
Thus,
\[ 1 + \cos^2(2A) = 2 - \sin^2(2A) \]

### Conclusion
The expression \(1 + \cos^2(2A)\) simplifies to:
\[ 2 - \sin^2(2A) \]