Question

Use a half-angle formula to rewrite the given expression so that it involves the sum or difference of only constants and sine and cosine functions to the first power.

(sin 3x)²(cos 3x)²

Answer 1

To simplify (sin 3x)²(cos 3x)², apply the half-angle formulas cos² θ = ½ (1 + cos 2θ) and sin² θ = ½ (1 - cos 2θ). Substituting these, the expression becomes ¼ (1 - cos² 6x), which simplifies to ¼ sin² 6x using the Pythagorean identity.

Step-by-step explanation:

To simplify the expression (sin 3x)²(cos 3x)² using half-angle formulas, we can make use of trigonometric identities. First, recall that the double angle formula for cosine is cos 2θ = cos² θ - sin² θ. We can rewrite this as cos 2θ = 2 cos² θ - 1 = 1 - 2 sin² θ. Solving for cos² θ and sin² θ, we get cos² θ = ½ (1 + cos 2θ) and sin² θ = ½ (1 - cos 2θ).

Using these identities, we can express (sin 3x)² and (cos 3x)² in terms of a double angle:

• (sin 3x)² = ½ (1 - cos 6x)
• (cos 3x)² = ½ (1 + cos 6x)

Substituting these into the original expression, we get:

(½ (1 - cos 6x)) (½ (1 + cos 6x)) = ¼ (1 - cos² 6x)

Finally, applying the Pythagorean identity sin² θ + cos² θ = 1, we can simplify further:

¼ (1 - (1 - sin² 6x)) = ¼ sin² 6x

Thus, the simplified expression in terms of sine and cosine functions is ¼ sin² 6x.