How do you simplify sin⁴x - cos⁴x ?

To simplify the expression sin 4x - cos 4x, we can use a trigonometric identity. One useful approach is to express it in terms of a single trigonometric function.

Using Trigonometric Identities

We can rewrite the expression by factoring it. Notice that:

• sin 4x can be expressed as sin(4x).
• cos 4x can be expressed as cos(4x).

Next, we can use the identity for the sine of a sum:

Factoring the Expression

We can factor out a common term:

sin 4x - cos 4x = √2 (sin 4x/√2 - cos 4x/√2)

This can be rewritten using the sine of a sum formula:

Final Simplification

Now, we can express it as:

√2 sin(4x - π/4)

Thus, the simplified form of sin 4x - cos 4x is:

√2 sin(4x - π/4)

This transformation makes it easier to analyze the function's behavior and graph it if needed.