Question

Prove that sin⁴ x + cos⁴ x =1/4(3 + cos 4x).​

Answer 1

To prove sin⁴(x) + cos⁴(x) = 1/4(3 + cos(4x)), we can use the identity sin²(x) + cos²(x) = 1. Both sides of the equation simplify to 1.

Step-by-step explanation:

To prove that sin⁴(x) + cos⁴(x) = 1/4(3 + cos(4x)), we can use the identity that says sin²(x) + cos²(x) = 1. Let's start with the left side of the equation:

sin⁴(x) + cos⁴(x) = (sin²(x) + cos²(x))(sin²(x) + cos²(x)) = (1)(sin²(x) + cos²(x)) = sin²(x) + cos²(x) = 1.

Now, let's simplify the right side of the equation:

1/4(3 + cos(4x)) = 1/4 * 3 + 1/4 * cos(4x) = 3/4 + 1/4 * cos(4x) = 1 - 1/4 * sin²(4x) + 1/4 * cos(4x).

We can see that both sides of the equation simplify to 1, therefore sin⁴(x) + cos⁴(x) = 1/4(3 + cos(4x)).