How do you prove cos⁴ (x) - sin⁴ (x) = cos (2x)?

sin(θ) is the ratio of the opposite (base) side to the hypotenuse, while cos(θ) is the ratio of the adjacent (perpendicular) side to the hypotenuse.

Answer: Hence proved that cos⁴ (x) - sin⁴ (x) = cos (2x)

Let's solve step by step using trigonometric identities.

Explanation:

We will start from LHS

cos⁴ (x) - sin⁴ (x) can be written as 

(cos² x)² - (sin² x)²

By using the property, a² - b² = (a + b) (a - b)

(cos² x)² - (sin² x)² = (cos² x + sin² x) (cos² x - sin² x)

= (1) × cos 2x, by using the trigonometric identities  

= cos 2x = RHS

So, LHS = RHS

Hence, it is proved that cos⁴ (x) - sin⁴ (x) = cos (2x)