Question #029c3

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Answer 1 · 2017-10-02T06:35:10

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

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.