Verify this trigonometrical identity?

Question

Show that

cos^4(x)-sin^4(x)=cos(2x)

Answer

We use the identities below.
(you should know them by heart)

sin^2(x) + cos^2(x) = 1 …(1)
cos(2x)= 2cos^2(x) – 1 …(2)

Put (2) into (1) from which we get,

cos(2x)
= 2cos^2(x) – [sin^2(x) + cos^2(x) ]

= cos^2(x) – sin^2(x) simplyfying

= [cos^2(x) – sin^2(x)]*[sin^2(x) + cos^2(x)] from (1)

= [cos^2(x) – sin^2(x)]*[cos^2(x) + sin^2(x)] rewriting

= [{cos^2(x)}^2 – {sin^2(x)}^2 ] by multiplying out

= cos^4(x) – sin^4(x)

Done.

We have shown the identity,

cos^4(x)-sin^4(x)=cos(2x)