How do you integrate $(1-(tanx)^2)/(secx)^2$ ?

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Answer 1 · 2016-12-28T22:46:14

I got $1/2sin(2x) + C$ .

Note the identity:

$1 + tan^2x = sec^2x$ ,

so that:

$int (1 - tan^2x)/(sec^2x)dx$

$int (2 - sec^2x)/(sec^2x)dx$

$int 2cos^2x - 1dx$

Now, note another identity:

$cos^2x = (1 + cos2x)/2$

Now this is much simpler to integrate than we started with!

$=> int 2((1 + cos2x)/2) - 1dx$

$= int cos2xdx$

Therefore:

$=> color(blue)(int (1 - tan^2x)/(sec^2x)dx = 1/2sin2x + C)$

How do you integrate (1-(tanx)^2)/(secx)^2(1-(tanx)^2)/(secx)^2?