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)
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