You should start considering the fundamental formula of trigonometry such that:
`(sin^2 A + cos^2 A) = 1`
Raising to square both sides yields:
`(sin^2 A + cos^2 A)^2 = 1^2`
`sin^4 A + 2sin^2A cos^2 A +cos^4 A = 1`
You need to isolate the sum `sin^4 A + cos^4 ` A to the left side such that:
`sin^4 A + cos^4 A = 1 - 2sin^2A cos^2 A`
You need to raise to square both sides such that:
`(sin^4 A + cos^4 A) = (1 - 2sin^2A cos^2 A)^2`
`sin^8 A + 2sin^4 Acos^4 A + cos^8 A = 1 - 4sin^2A cos^2 A + 4sin^4 A cos^4 A`
Moving to the right `2sin^4 Acos^4 A` yields:
`sin^8 A +cos^8 A = 1 - 4sin^2A cos^2 A + 4sin^4 A cos^4 A - 2sin^4 Acos^4 ` A
`sin^8 A + cos^8 A = 1 - 4sin^2A cos^2 A + 2sin^4 A cos^4 A`
Hence, checking if the given identity holds yields that, using the fundamental formula of trigonometry, the identity `sin^8 A + cos^8 A = 1 - 4sin^2A cos^2 A + 2sin^4 A cos^4 A` is valid.