Prove that: 1/(1 + sin x) = sec x(sec x - tan x)

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We have to prove that: 1/(1 + sin x) = sec x(sec x - tan x)

Start with the right hand side:

sec x(sec x - tan x)

=> `sec^2 x - (sec x)*(tan x)`

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

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

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

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

=> `1/(1+sin x)`

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

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