1/ sec x - tan x …
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Sia ? 6 years, 6 months ago
LHS = {tex}\frac{1}{sec x - tan x}{/tex} - {tex}\frac{1}{cos x}{/tex}
={tex}\frac{sec x + tan x}{(sec x - tan x)(sec x + tan x)}{/tex} - {tex}\frac{1}{cos x}{/tex}
= {tex}\frac{secx + tan x}{sec^2 x - tan^2 x}{/tex} - {tex}\frac{1}{cos x}{/tex}
{tex}=sec x + tan x - sec x{/tex}{tex}[\because sec^2\theta-tan^2\theta=1]{/tex}
= tan x
RHS = {tex}\frac{1}{cosx - 1}{/tex} - {tex}\frac{1}{sec x + tan x}{/tex}
={tex}\frac{1}{cos x}{/tex} - {tex}\frac{sec x - tan x}{(sec x + tan x)(sec x - tan x)}{/tex}
={tex}\frac{1}{cos x}{/tex} - {tex}\frac{sec x - tan x}{sec^2 x - tan^2 x}{/tex}
= sec x - {tex}\frac{sec x - tan x}{1}{/tex}
{tex}=sec x - sec x + tan x{/tex}
{tex}=tan x{/tex}
Hence, LHS = RHS
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