Question

In exercises 5 and 7, tell whether or not f(x) = sin x is an identity.

5. F(x) = sin^2 + cos^2x/csc x

7. F(x) = (cos x)(cot x)

Answer 1

For 5, we can make 1/csc x = sinx, but we're then left with sin^2+cos^2x(sinx)=sin(sin+cos^2x), which doesn't give us anything. False.

For 7, we know that cosx= tanx/sinx and cotx=1/tanx, so we cross out the tan x's and get 1/sinx, which is not sinx. False

15 - (1-cos^2u)/cos^2u=tan^2u, turn 1-cos^2u=sin^2u and square root both sides to get sin/cos=tan

17 - tanx = sinx/cosx, so multiply that with sinx on the right to get sin^2x/cosx, and multiply both sides by cosx to get cos^2x-1=sin^2x (assuming that (cos^2x-1)/cosx is what was meant on the right)

23 - don't know how to prove that true, sorry

31 - (cos^4x-sin^4x)=(cosx+sinx)(cosx-sinx)(cos^2x+sin^2x)=(cosx+sinx)(cosx-sinx)=cos^2x-sin^2x