Prove the identity: (cos x / (1-sin x)) - tan x = 1/cos x

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Prove the identity `cosx/((1-sinx))-tanx=1/cosx` :

`cosx/((1-sinx))-tanx`

`=cosx/((1-sinx))-sinx/cosx` trig substitution identity

`=(cos^2x-sin(x)(1-sinx))/(cosx(1-sinx))` Add fractions (common denominator)

`=(cos^2x+sin^2x-sinx)/(cosx(1-sinx))` Distributive property;rearrange terms

`=(1-sinx)/(cosx(1-sinx))` Pythagorean identity

`=1/cosx` Cancel like factors

QED.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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It has to be proven that `(cos x / (1 - sin x)) - tan x = 1/cos x`

`(cos x / (1 - sin x)) - tan x`

=> `(cos x / (1 - sin x)) - (sin x/cos x)`

=> `(cos^2x - (1 - sin x)*sin x)/((1 - sin x)*cos x)`

=> `(cos^2x - sin x + sin^2 x)/((1 - sin x)*cos x)`

=> `(1 - sin x)/((1 - sin x)*cos x)`

=> `1/cos x`

This proves that `(cos x / (1 - sin x)) - tan x = 1/cos x`

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