Final answer:
To simplify the expression 2sin3θ cos3θ, we can use the double-angle formula for sine and cosine. The simplified form of the expression is 4sinθ cosθ (cos2θ - sin2θ).
Step-by-step explanation:
To simplify the expression 2sin3θ cos3θ, we can use the double-angle formula for sine. The double-angle formula for sine states that sin2θ = 2sinθ cosθ. Applying this formula, we can rewrite sin3θ as 2sinθ cosθ. Substituting this into the given expression, we get 2(2sinθ cosθ) cos3θ.
Next, we can use the double-angle formula for cosine. The double-angle formula for cosine states that cos2θ = cos2θ - sin2θ. Applying this formula, we can rewrite cos3θ as cos2θ - sin2θ. Substituting this into the expression, we have 2(2sinθ cosθ) (cos2θ - sin2θ).
Further simplifying, we get 4sinθ cosθ (cos2θ - sin2θ). This is the simplified form of the expression.