Prove the following identities: (i)cos^2x (1+tan^2x)=1, (ii) (2 sin x +cos x)^2=5, (iii) (1+sin x/cos x)^2=1+sin x/1-sin x, (iv) 1- (sin^2 x)/1+cos x= cos x, (v) 1-2cos^2 x/sin x+cos x= sinx-cosx.

Answer 1

First thing is to write out the identities:

1. sin^2(x) +cos^2(x) =1
2. 1 + tan^2(x)=sec^2(x)
3. 1 + cot^2(x)=csc^2(x)
4. We know that Replace the second term with sec^2(x). Secant is a reciprocal of cosine, so we have cos^2(x)*1/cos^2(x)=1. Identity proven.

The rest can be simplified to make one side equal the other using a combination of identities.