Final answer:
The trigonometric identity can be proven using the Pythagorean identity and algebraic manipulation. By expressing sin^6x and cos^6x as cubes of sin^2x and cos^2x respectively, and then applying the binomial expansion, the left side simplifies to match the right side.
Step-by-step explanation:
To prove the identity cos^6x + sin^6x = 5/8 + 3/8cos^4x, we can use trigonometric identities to manipulate the left side of the equation to match the right side. First, recall the Pythagorean identity sin^2x + cos^2x = 1. Next, notice that both sin^6x and cos^6x can be written as (sin^2x)^3 and (cos^2x)^3 respectively. Now, we apply the formula for the cube of a sum: (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3. Using the substitution a = cos^2x and b = sin^2x, and taking into account that a + b = 1 (from the Pythagorean identity), we can rewrite the original expression as:
(cos^2x + sin^2x)^3 - 3cos^2x sin^2x(cos^2x + sin^2x)
This simplifies to 1^3 - 3cos^2x sin^2x, and since cos^2x sin^2x = (1 - cos^2x)(cos^2x) (another use of the Pythagorean identity), we further simplify it to:
1 - 3cos^2x + 3cos^4x
Expanding and simplifying using algebra, we get:
1 - 3/4 + 3/8
Finally, combine like terms to match the right side of the original equation, which completes our proof.