This page covers the derivation and use of the tangent reduction formula for integration.
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Tangent Reduction Formula (n is an integer and \(n>1\)) 

\(\displaystyle{\int{\tan^n x~dx}= \frac{\tan^{n1}x}{n1}  \int{\tan^{n2}x~dx}}\) 
When you have an integral with only tangent where the power is greater than one, you can use the tangent reduction formula, repeatedly if necessary, to reduce the power until you end up with either \(\tan x\) or \(\tan^2 x\). Let's derive the formula and then work some practice problems.
Deriving The Tangent Reduction Formula
Separate out a \(\tan^2x\). 
\(\int{\tan^nx~dx} = \int{\tan^{n2}x \tan^2x~dx}\) 
Use the identity \(\sec^2x=1+\tan^2x\) to replace the \(\tan^2x\) with \(\sec^2x1\). 
\(\int{\tan^nx~dx} = \int{\tan^{n2}x~(\sec^2x1)~dx}\) 
Distribute the \(\tan^{n2}x\) term and separate into two integrals. 
\(\int{\tan^nx~dx} =\) \(\int{\tan^{n2}x \sec^2x~dx}\) \(\) \(\int{\tan^{n2}x~dx}\) 
Use integration by substitution on the first integral on the right side of the equal sign. 
\(u=\tan x \to du=\sec^2x~dx\) 
\(\int{\tan^nx~dx} =\) \(\int{u^{n2}~du}\) \(\) \(\int{\tan^{n2}x~dx}\) 
Integrate the term containing u and convert back to x's. 
\(\displaystyle{ \int{\tan^nx~dx} = }\) \(\displaystyle{ \frac{u^{n1}}{n1}  \int{\tan^{n2}x~dx} }\) 
\(\displaystyle{ \int{\tan^nx~dx} = \frac{\tan^{n1}x}{n1}  \int{\tan^{n2}x~dx} }\) 

This last equation is the tangent reduction formula. 
Now let's work some practice problems.
Practice
Unless otherwise instructed, evaluate these integrals directly, then check your answer using the reduction formula.
\(\displaystyle{ \int{ \tan^3x ~dx } }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ \tan^3x ~dx } }\) using the techniques on the tangentsecant trig integration page. Then check your answer using the tangent reduction formula.
Solution
video by The Organic Chemistry Tutor 

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\(\displaystyle{ \int{ \tan^4x ~dx } }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ \tan^4x ~dx } }\) using the techniques on the tangentsecant trig integration page. Then check your answer using the tangent reduction formula.
Solution
video by The Organic Chemistry Tutor 

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Evaluate \( \int{ \tan^5x ~dx } \) using the tangent reduction formula.
Problem Statement
Evaluate \( \int{ \tan^5x ~dx } \) using the tangent reduction formula.
Solution
Here are two videos, by two different instructors, solving this problem.
video by PatrickJMT 

video by The Organic Chemistry Tutor 

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Really UNDERSTAND Calculus
basic trig identities 

\(\sin^2\theta+\cos^2\theta=1\)  \(1+\tan^2\theta=\sec^2\theta\) 
\(\displaystyle{\tan\theta=\frac{\sin\theta}{\cos\theta}}\)  \(\displaystyle{\cot\theta=\frac{\cos\theta}{\sin\theta}}\) 
\(\displaystyle{\sec\theta=\frac{1}{\cos\theta}}\)  \(\displaystyle{\csc\theta=\frac{1}{\sin\theta}}\) 
power reduction (halfangle) formulae 
\(\displaystyle{\sin^2\theta=\frac{1\cos(2\theta)}{2}}\)  \(\displaystyle{\cos^2\theta=\frac{1+\cos(2\theta)}{2}}\) 
double angle formulae 
\(\sin(2\theta)=2\sin\theta\cos\theta\)  \(\cos(2\theta)=\cos^2\theta\sin^2\theta\) 
links 
basic trig derivatives  

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) 
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = \sin(t) }\)  
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) 
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = \csc^2(t) }\)  
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = }\) \(\sec(t)\tan(t) \) 
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = }\) \( \csc(t)\cot(t) \)  
basic trig integrals  
\(\int{\sin(x)~dx} = \cos(x)+C\)  
\(\int{\cos(x)~dx} = \sin(x)+C\)  
\(\int{\tan(x)~dx} = \ln\abs{\cos(x)}+C\)  
\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)  
\(\int{\sec(x)~dx} = \ln\abs{\sec(x)+\tan(x)}+C\)  
\(\int{\csc(x)~dx} = \ln\abs{\csc(x)+\cot(x)}+C\)  
reduction formulae  
Reduction Formulas (n is a positive integer)  
\(\displaystyle{\int{\sin^n x~dx} = \frac{\sin^{n1}x\cos x}{n} + }\) \(\displaystyle{ \frac{n1}{n}\int{\sin^{n2}x~dx} }\)  
\(\displaystyle{\int{\cos^n x~dx} = \frac{\cos^{n1}x\sin x}{n} + }\) \(\displaystyle{ \frac{n1}{n}\int{\cos^{n2}x~dx}}\)  
Reduction Formulas (n is an integer and \(n>1\))  
\(\displaystyle{\int{\tan^n x~dx}= \frac{\tan^{n1}x}{n1}  \int{\tan^{n2}x~dx}}\)  
\(\displaystyle{\int{\sec^n x~dx} = \frac{\sec^{n2}x\tan x}{n1} + }\) \(\displaystyle{ \frac{n2}{n1}\int{\sec^{n2}x~dx}}\)  
links  
related topics on other pages 

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Practice Instructions
Unless otherwise instructed, evaluate these integrals directly, then check your answer using the reduction formula.