How do you verify cos^4(x)-sin^4(x)=cos2x using the double angle identity?

2 Answers
Aug 9, 2015

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

Explanation:

Factor the left hand side. It is the difference of two squares.

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

From the Pythagorean identity we know that

cos^2x+sin^2x=1 so we can write

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

Now

cos(2x)=cos(x+x)

From the angle sum identity we have

cos(2x)=cos(x+x)=cos(x)cos(x)-sin(x)sin(x)

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

Which shows that the original equation is true

Aug 10, 2015

Simplify cos^4 x - sin^4 x

Explanation:

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

Reminder of 2 trig identities:
cos^2 x + sin^2 x = 1
cos^2 x - sin^2 x = cos 2x