Prove that  sin⁶ A + cos⁶ A = 1 - 3sin² A cos² A

Make use of the trigonometric identity as well as the cubic formula to answer this question. 

Answer: Making both the sides equal in the given solution, proves the desired expression.

We will prove this identity using trigonometric identities.

Explanation:

We can make use of cubic formulas to deduce the expression into a simplified form.

Given, LHS = sin⁶ A + cos⁶ A 

Using the rule, a⁶ = (a²)³, the above expression is reduced to,

= (sin² A)³ + (cos² A)³ 

= (sin² A + cos² A)³ - 3 sin² A cos² A  

= 1³ - 3 sin² A cos² A

= 1 - 3 sin² A cos² A

LHS = RHS

Hence Proved that sin⁶ A + cos⁶ A = 1 - 3 sin² A cos² A.