# Integrate sin(2x)tan(2x)

## \int\sin\left(2x\right)\cdot\tan\left(2x\right)dx

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$4\int\frac{\cos\left(x\right)^2\sin\left(x\right)^2}{\cos\left(2x\right)}dx$

## Step by step solution

Problem

$\int\sin\left(2x\right)\cdot\tan\left(2x\right)dx$
1

Applying the tangent identity: $\displaystyle\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$

$\int\frac{\sin\left(2x\right)}{\cos\left(2x\right)}\sin\left(2x\right)dx$
2

Using the sine double-angle identity

$\int\frac{2\cos\left(x\right)\sin\left(x\right)}{\cos\left(2x\right)}\sin\left(2x\right)dx$
3

Multiplying the fraction and term

$\int\frac{2\sin\left(2x\right)\cos\left(x\right)\sin\left(x\right)}{\cos\left(2x\right)}dx$
4

Using the sine double-angle identity

$\int\frac{4\cos\left(x\right)\sin\left(x\right)\cos\left(x\right)\sin\left(x\right)}{\cos\left(2x\right)}dx$
5

When multiplying exponents with same base you can add the exponents

$\int\frac{4\cos\left(x\right)^2\sin\left(x\right)^2}{\cos\left(2x\right)}dx$
6

Taking the constant out of the integral

$4\int\frac{\cos\left(x\right)^2\sin\left(x\right)^2}{\cos\left(2x\right)}dx$
7

Applying the trigonometric identity: $\sin^2(\theta)=1-\cos(\theta)^2$

$4\int\frac{\cos\left(x\right)^2\left(1-\cos\left(x\right)^2\right)}{\cos\left(2x\right)}dx$
8

Applying the trigonometric identity: $1-\cos\left(\theta\right)^2=\sin\left(\theta\right)^2$

$4\int\frac{\cos\left(x\right)^2\sin\left(x\right)^2}{\cos\left(2x\right)}dx$

$4\int\frac{\cos\left(x\right)^2\sin\left(x\right)^2}{\cos\left(2x\right)}dx$

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### Main topic:

Integral calculus

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