Question

Question #029c3

Answer 1

Proved `1/[cos x(1+cos x)]= [tan x -sin x]/sin^3x`

Explanation:

Given, `1/[cos x(1+cos x)] = [tan x - sin x]/sin^3x`

We have to prove either way. Let me take Left Hand Side (L.H.S.)

`1/[cos x (1+cos x)]`

`rArr [1.(1-cos x)]/[cos x (1+cos x)(1-cos x)` [ multiply both sides by (1 - cos x)]

`rArr [1 - cos x]/[cos x (1 - cos^2 x)]`

`rArr [(1-cos x)/cos x]/sin^2x` [ as `sin^2x + cos^2x = 1. so, 1-cos^2x = sin^2x]`

`rArr [1/cos x - cos x/cos x]/sin^2x`

`rArr [(1/cos x -1)sin x]/[sin^2 x. sin x]` [ multiply both sides by sin x]

`rArr [sin x/cos x - sin x]/sin^3x`

`rArr (tan x - sin x)/sin^3x` = R. H. S.