How do you prove $sec^2 x - cot^2 ( pi/2-x) =1$ ?

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Answer 1 · 2016-02-05T22:32:46

Using the following:

We have

$sec^2(x)-cot^2(pi/2-x) = sec^2(x) - cot^2(-(x-pi/2))$

$=1/cos^2(x) - cos^2(-(x-pi/2))/sin^2(-(x-pi/2))$

$=1/cos^2(x)-cos^2(x-pi/2)/(-sin(x-pi/2))$

$=1/cos^2(x) - sin^2(x)/cos^2(x)$

$= (1-sin^2(x))/cos^2(x)$

$=cos^2(x)/cos^2(x)$

$= 1$

How do you prove sec^2 x - cot^2 ( pi/2-x) =1sec^2 x - cot^2 ( pi/2-x) =1?