If sin x + cos x = a, find the value of sin⁶ x + cos⁶ x.

Given, sin x + cos x = a

We need to find the value of the expression,

sin⁶ x + cos⁶ x = (sin² x)³ + (cos² x)³

= (sin² x + cos² x)³ – 3 sin² x cos² x (sin² x + cos² x)

[ by using the formula a³ + b³ = (a+b)³ – 3ab(a+b)]

= (1)³ – 3 sin² x cos² x (1)

[ by using the formula sin² x + cos² x = 1]

[ by using the formula sin² x + cos² x = 1]

Hence sin⁶ x + cos⁶ x