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.