Verify the identity (sinx + sin2x)/(1+cosx + cos2x)= tan x Please show step by step instructions

Expert Answers

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You should use the double angle formulas such that:

`sin 2x = 2 sin x cos x`

`cos 2x = 2cos^2 x - 1`

Hence, substituting `2 sin x cos x`  for `sin 2x`  and `2cos^2 x - 1`  for `cos 2x`  yields:

`(sinx + 2 sin x cos x)/(1+ cosx + 2cos^2 x - 1)= tan x`

You need to factor out sin x to numerator and cos x to denominator such that:

`(sin x(1 + 2 cos x))/(cos x(1 + 2 cos x)) = tan x`

Reducing `by 1+ 2 cos x`  yields:

`sin x/cos x = tan x => tan x = tan x`

Hence, using the double angle formulas, yields that the given identity `(sinx + sin2x)/(1+cosx + cos2x)= tan x`  holds.

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