tan2A+tan3A+tan2Atan3Atan5A=

I'm guessing those are coefficients of A.

Use:

\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A\tan B}

I used tan2Atan3Atan5A=tan5A-tan3A-tan2A... is this method correct??

I don't know this identity, but it's valid :)