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Trig Identities and Formulas

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 (half-angle) 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

list of trigonometric identities - wikipedia

trig sheets - pauls online notes

17calculus trig formulas - full list

Trig Derivatives and Integrals

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)

\(\displaystyle{\int{\sin^n x~dx} = -\frac{\sin^{n-1}x\cos x}{n} + }\) \(\displaystyle{ \frac{n-1}{n}\int{\sin^{n-2}x~dx} }\)

\(\displaystyle{\int{\cos^n x~dx} = \frac{\cos^{n-1}x\sin x}{n} + }\) \(\displaystyle{ \frac{n-1}{n}\int{\cos^{n-2}x~dx}}\)

Reduction Formulas (n is an integer and \(n>1\))

\(\displaystyle{\int{\tan^n x~dx}= \frac{\tan^{n-1}x}{n-1} - \int{\tan^{n-2}x~dx}}\)

\(\displaystyle{\int{\sec^n x~dx} = \frac{\sec^{n-2}x\tan x}{n-1} + }\) \(\displaystyle{ \frac{n-2}{n-1}\int{\sec^{n-2}x~dx}}\)

links

17calculus trig formulas - full list

Related Topics and Links

17Calculus Subjects Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

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This page covers the derivation and use of the tangent reduction formula for integration.

Tangent Reduction Formulas (n is an integer and \(n>1\))

\(\displaystyle{\int{\tan^n x~dx}= \frac{\tan^{n-1}x}{n-1} - \int{\tan^{n-2}x~dx}}\)

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 \(\tan^2x\).

\(\int{\tan^nx~dx} = \int{\tan^{n-2}x \tan^2x~dx}\)

Use the identity \(\sec^2x=1+\tan^2x\) to replace the \(\tan^2x\) with \(\sec^2x-1\).

\(\int{\tan^nx~dx} = \int{\tan^{n-2}x~(\sec^2x-1)~dx}\)

Distribute the \(\tan^{n-2}x\) term and separate into two integrals.

\(\int{\tan^nx~dx} =\) \(\int{\tan^{n-2}x \sec^2x~dx}\) \(-\) \(\int{\tan^{n-2}x~dx}\)

Use integration by substitution on the first integral on the right side of the equal sign.

\(u=\tan x \to du=\sec^2x~dx\)

\(\int{\tan^nx~dx} =\) \(\int{u^{n-2}~du}\) \(-\) \(\int{\tan^{n-2}x~dx}\)

Integrate the term containing u and convert back to x's.

\(\displaystyle{ \int{\tan^nx~dx} = }\) \(\displaystyle{ \frac{u^{n-1}}{n-1} - \int{\tan^{n-2}x~dx} }\)

\(\displaystyle{ \int{\tan^nx~dx} = \frac{\tan^{n-1}x}{n-1} - \int{\tan^{n-2}x~dx} }\)

This last equation is the tangent reduction formula.

Now let's work some practice problems.

Practice

\(\displaystyle{ \int{ \tan^5x ~dx } }\)

Problem Statement

\(\displaystyle{ \int{ \tan^5x ~dx } }\)

Solution

116 solution video

video by PatrickJMT

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