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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