Trigonometry(Compound Angles)

Question

Prove that:- $$\tan 2 \theta+\tan 3 \theta -\tan5\theta=-\tan 2\theta.\tan3\theta.\tan5\theta$$

I cannot figure out how to start solving this problem......

Answer 1

You can solve your problem if you use $$\tan (x+y) = {\tan x+ \tan y \over 1-\tan x\cdot \tan y}$$

Answer 2

lhs:

$tan2\theta+tan3\theta-tan5\theta$

using the formula with $x=2\theta$ and $y=3\theta$

($tan5\theta$)($1-tan2\theta tan3\theta)$ - $tan5\theta$

multiplying out gives

$tan5\theta-tan2\theta tan3\theta tan5\theta-tan5\theta$

equals the rhs

so the question should read

$tan2\theta+tan3\theta-tan5\theta = -tan2\theta tan3\theta tan5\theta$