Use the identities cos^2 x + sin^2 x =1
and
cos2x=cos^2 x -sin^2 x
to show that
cos^4 x -sin^4 x = cos2x
Im not sure how, I can solve my problem with half angle identities but im not sure where to start with this.
use x^2-y^2 = (x+y)(x-y)
c^4-s^4 = (c^2+s^2)(c^2-s^2)
= 1 (c^2-s^2)
= c 2x
To prove that cos^4 x - sin^4 x = cos2x using the given identities, we will break down each side of the equation one step at a time:
Starting with the left-hand side (LHS):
cos^4 x - sin^4 x
We can rewrite this expression using the difference of squares formula:
(cos^2 x + sin^2 x)(cos^2 x - sin^2 x)
Since we know that cos^2 x + sin^2 x is equal to 1 (from the identity cos^2 x + sin^2 x = 1), we can substitute this value:
(1)(cos^2 x - sin^2 x)
Now, let's use the identity cos2x = cos^2 x - sin^2 x:
(1)(cos2x)
Simplifying, we get:
cos2x
Therefore, we have shown that cos^4 x - sin^4 x = cos2x using the given identities.