`cos^(4)theta-sin^(4)theta` is :

Options

A. `cos 2 theta`
B. `sin2 theta`
C. `tan2theta`
D. none of these

Answer

To solve the expression \( \cos^4 \theta - \sin^4 \theta \), we can use the difference of squares identity. Here’s the step-by-step solution:

### Step 1: Recognize the Difference of Squares
The expression \( \cos^4 \theta - \sin^4 \theta \) can be recognized as a difference of squares. We can rewrite it as:
\[ \cos^4 \theta - \sin^4 \theta = (\cos^2 \theta)^2 - (\sin^2 \theta)^2 \]

### Step 2: Apply the Difference of Squares Formula
Using the difference of squares formula \( A^2 - B^2 = (A - B)(A + B) \), where \( A = \cos^2 \theta \) and \( B = \sin^2 \theta \), we can factor the expression:
\[ \cos^4 \theta - \sin^4 \theta = (\cos^2 \theta - \sin^2 \theta)(\cos^2 \theta + \sin^2 \theta) \]

### Step 3: Simplify Using Trigonometric Identities
We know from the Pythagorean identity that:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]
Substituting this into our expression gives:
\[ \cos^4 \theta - \sin^4 \theta = (\cos^2 \theta - \sin^2 \theta)(1) \]
Thus, we have:
\[ \cos^4 \theta - \sin^4 \theta = \cos^2 \theta - \sin^2 \theta \]

### Step 4: Further Simplification
The expression \( \cos^2 \theta - \sin^2 \theta \) can be rewritten using the double angle formula:
\[ \cos^2 \theta - \sin^2 \theta = \cos(2\theta) \]

### Final Result
Therefore, the expression \( \cos^4 \theta - \sin^4 \theta \) simplifies to:
\[ \cos^4 \theta - \sin^4 \theta = \cos(2\theta) \]

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