Prove that: sin⁶ x + cos⁶ x = 1 - 3 sin² x cos² x.

Cosine and Sine are primary trigonometric functions. They are also known as complemetary functions

Answer: sin⁶ x + cos⁶ x = 1 - 3 sin² x cos² x

To solve this, we will use algebraic identity.

Explanation:

sin⁶ x + cos⁶ x  can be written as (sin²x )³ + (cos²x )³

Using Algebraic identity,

a³ + b³ = (a + b)³ - 3ab (a + b) 

(sin²x )³ + (cos²x )³ = (sin²x + cos²x )³  - 3 sin²x cos²x (sin²x + cos²x )

Using trigonometric identity sin²x + cos²x = 1, we get

⇒ sin⁶ x + cos⁶ x  = (1) ³ - 3 sin²x cos²x (1)

⇒ sin⁶ x + cos⁶ x  = 1 - 3 sin²x cos²x 

Thus, proved that sin⁶ x + cos⁶ x = 1 - 3 sin² x cos² x