Question

(1-cosx)/(tanx) + (sinx)/(1+cosx) trig identity’s

Answer 1

sinx

Explanation:

(1 -cosx)/(tanx) + (sinx)/(1 +cosx) =

-add the fructions using the common denominator

[(1- cosx)( 1+ cosx) + (sin x) ( tan x) ] / (1+ cos x) (tanx) =

-use that: a²-b²= (a+b) (a-b) so (1- cosx)( 1+ cosx) = 1 - cos²x

and that tanx = sin x/ cosx

[(1- cos²x) + (sinx) ( sinx/cosx) ] / (1+ cos x) (tanx) = =

use that: sin²x+cos²x = 1, so sin²x = 1 - cos²x and then add the fractions

[(sin²x cosx + sin²x)/ cos x ] ·(1/ (1+ cos x) (tanx) ] =

[(sinx (sinx cosx + sinx)/ cos x ] ·(1/ (1+ cos x) (tanx) ] =

[tanx (sinx cosx + sinx) / (1+ cos x) (tanx) ] =

sinx (cosx + 1) / (1+ cos x) =

sinx