How can you simplify sin^4a + cos^4asin4a+cos4a?

1 Answer
Jul 17, 2017

sin^4a + cos^4a = 1 - 2sin^2acos^2a = 1 - 1/2sin^2(2a)sin4a+cos4a=12sin2acos2a=112sin2(2a).

Explanation:

Note that

(sin^2a + cos^2a)^2 = sin^4a + 2sin^2acos^2a + cos^4a(sin2a+cos2a)2=sin4a+2sin2acos2a+cos4a

We see that sin^2x + cos^2x = 1sin2x+cos2x=1, so that

1 = sin^4a + 2sin^2acos^2a + cos^4a1=sin4a+2sin2acos2a+cos4a

Therefore,

sin^4a + cos^4a = 1 - 2sin^2acos^2asin4a+cos4a=12sin2acos2a

Hopefully this helps!