Prove the identity

Question

Cos⁴x + sin²x = sin⁴x + cos²x

Answer 1

We can start by manipulating the left side of the equation:

cos⁴x + sin²x = (cos²x)² + sin²x
= (cos²x)² + (1 - cos²x)
= (cos²x)² + 1 - cos²x
= (cos²x)² - cos²x + 1.

Now let's manipulate the right side of the equation:

sin⁴x + cos²x = (1 - cos²x)² + cos²x
= (1 - 2cos²x + (cos²x)²) + cos²x
= 1 - 2cos²x + (cos²x)² + cos²x
= (cos²x)² - 2cos²x + 1 + cos²x
= (cos²x)² - cos²x + 1.

Therefore, we have shown that the left side of the equation is equal to the right side of the equation, proving the identity:

cos⁴x + sin²x = sin⁴x + cos²x.