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.
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.