How do you prove that $cosx/(1-sinx) - 1/cosx = tanx$ ?

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Answer 1 · 2015-04-16T16:06:36

$f(x) = cos x/(1 - sin x) - 1/cos x = [cos ^2 x - (1 - sin x)]/(cos x*(1 - sin x)$

In the numerator of $f(x),$ replace

$cos^2 x = 1 - sin^2 x$

Numerator $= 1 - sin^2 x - 1 + sin x =$

$sin x*(1 - sin x)$

$f(x) = (sin x*(1 - sin x))/(cos x*(1 - sin x))= sin x/cos x = tan x.$

How do you prove that cosx/(1-sinx) - 1/cosx = tanxcosx/(1-sinx) - 1/cosx = tanx?