What is the integral of (1-tan^2(x))/(sin^2(x)) ?

The integral of (1-tan^2(x))/(sin^2(x)) is -cot(x)-tan(x)+C

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integral of (1-tan^2(x))/(sin^2(x))

\int \frac{1 - tan ^{2} ( x )}{sin ^{2} ( x )} d x

- cot (x) - tan (x) + C

Solution steps

\int \frac{1 - tan ^{2} ( x )}{sin ^{2} ( x )} d x

Expand

\frac{1 - tan ^{2} ( x )}{sin ^{2} ( x )} : \frac{1}{sin ^{2} ( x )} - \frac{tan ^{2} ( x )}{sin ^{2} ( x )}

= \int \frac{1}{sin ^{2} ( x )} - \frac{tan ^{2} ( x )}{sin ^{2} ( x )} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{sin ^{2} ( x )} d x - \int \frac{tan ^{2} ( x )}{sin ^{2} ( x )} d x

\int \frac{1}{sin ^{2} ( x )} d x = - cot (x)

\int \frac{tan ^{2} ( x )}{sin ^{2} ( x )} d x = tan (x)

= - cot (x) - tan (x)

Add a constant to the solution

= - cot (x) - tan (x) + C

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What is the integral of (1-tan^2(x))/(sin^2(x)) ?
The integral of (1-tan^2(x))/(sin^2(x)) is -cot(x)-tan(x)+C