Question

Verify the identity: sin⁴ 4x−4sin² 2xcos² 2x=0.

A) True
B) False
C) Cannot be determined
D) None of the above

Answer 1

The identity sin⁴ 4x − 4sin² 2x cos² 2x = 0 is verified to be true by using trigonometric identities and the double angle identity, demonstrating that both terms in the equation cancel each other out.

Step-by-step explanation:

To verify the identity sin⁴ 4x − 4sin² 2x cos² 2x = 0, we need to utilize trigonometric identities. Firstly, apply the double angle identity sin 2θ = 2 sin θ cos θ to express sin 4x in terms of sin 2x and cos 2x.

Consider sin 2θ = 2 sin θ cos θ and let θ = 2x:

• sin 4x = 2 sin 2x cos 2x

Raising both sides to the second power:

• (sin 4x)² = (2 sin 2x cos 2x)²
• sin⁴ 4x = 4 sin² 2x cos² 2x

Substitute sin⁴ 4x in the original equation:

• 4 sin² 2x cos² 2x − 4 sin² 2x cos² 2x = 0

As you can see, both terms are identical and they cancel each other out, yielding zero, which verifies that the given identity is indeed true.