prove the following trig identity

Question

sin^4x - cos^4x =1 -2cos^2x

Answer 1

sin⁴θ - cos⁴θ =1 - 2cos²θ

sin⁴θ-cos⁴θ is the difference of two squares =(sin²θ-cos²θ)(sin²θ+cos²θ)=(sin²θ-cos²θ)1

sin²θ-cos²θ=1-cos²θ-cos²θ = 1 - 2cos²θ

Answer 2

sin⁴θ - cos⁴θ =1 - 2cos²θ

 

use the identity: sin²θ = 1 - cos²θ

 

start with the left side

 

sin⁴θ - cos⁴θ

 

(sin²θ)² - cos⁴θ

 

(1 - cos²θ)(1 - cos²θ) -  cos⁴θ

 

expand

 

1 - 2cos²θ + cos⁴θ - cos⁴θ

 

1 - 2cos²θ