What is the integral of (cos^2(θ))/(sin^2(θ)) ?

The integral of (cos^2(θ))/(sin^2(θ)) is -cot(θ)-θ+C

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integral of (cos^2(θ))/(sin^2(θ))

\int \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )} d θ

- cot (θ) - θ + C

Solution steps

\int \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )} d θ

Rewrite using trig identities

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

Expand

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

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

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} ( θ )} d θ - \int 1 d θ

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

\int 1 d θ = θ

= - cot (θ) - θ

Add a constant to the solution

= - cot (θ) - θ + C

What is the integral of (cos^2(θ))/(sin^2(θ)) ?
The integral of (cos^2(θ))/(sin^2(θ)) is -cot(θ)-θ+C