Question

Which of the following are identities? More than 1 answer:

a. sin x+ sin5x=tan3x
b. sin^6 x+cos^6 x=4-3sin^2 2x
c. cos^2 3x-sin^2 3x=cos6x
d.(sin x+ cos x)^2 =1+sin 2x

Answer 1

Answer: C and D

Explanation:

Answer 2

In order to confirm which of the given above is an identity, what we are going to do is to check them each. By definition, an identity is an equality relation A = B.
After checking each options, the answers that are considered as identities would be options C and D. So here is how we proved it. Let's take option C.
cos^2(3x)-sin^2(3x)=cos(6x)
cos^2(3x)-sin^2(3x)=cos(2*3x)
cos^2(3x)-sin^2(3x)=cos(3x+3x)
cos^2(3x)-sin^2(3x)=cos(3x)cos(3x)-sin(3x)sin(3x)
cos^2(3x)-sin^2(3x)=cos^2(3x)-sin^2(3x)
So based on this, we can conclude that cos^2 3x-sin^2 3x=cos6x is an identity.
This is also the same process with option D.
Hope this answer helps.