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Trig Identities and Formulas for Derivatives and Integrals 

on this page: ► basic trig identities ► trig derivatives ► trig integrals ► inverse trig derivatives 
This page combines formulas found on other pages including basic identities, derivatives and integrals for trig and inverse trig functions. We also include some trig reduction formulas used for trig integration. 
Basic Trig Identities 

Set 1  basic identities  

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\) 
\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\) 
\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\) 
\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\) 
Set 2  squared identities  

\( \sin^2t + \cos^2t = 1\) 
\( 1 + \tan^2t = \sec^2t\) 
\( 1 + \cot^2t = \csc^2t\) 
Set 3  doubleangle formulas  

\( \sin(2t) = 2\sin(t)\cos(t)\) 
\(\displaystyle{ \cos(2t) = \cos^2(t)  \sin^2(t) }\) 
Set 4  halfangle formulas  

\(\displaystyle{ \sin^2(t) = \frac{1\cos(2t)}{2} }\) 
\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\) 
Trig Derivatives 

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) 
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = \sin(t) }\)  
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) 
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = \csc^2(t) }\)  
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\) 
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = \csc(t)\cot(t) }\) 
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 Formulas (where n is a positive integer)  

\(\displaystyle{\int{\sin^n x~dx} = \frac{\sin^{n1}x\cos x}{n}+\frac{n1}{n}\int{\sin^{n2}x~dx}}\) 

\(\displaystyle{\int{\cos^n x~dx} = \frac{\cos^{n1}x\sin x}{n} + \frac{n1}{n}\int{\cos^{n2}x~dx}}\) 
Reduction Formulas (where n is an integer and \(n>1\))  

\(\displaystyle{\int{\tan^n x~dx}= \frac{\tan^{n1}x}{n1}  \int{\tan^{n2}x~dx}}\) 

\(\displaystyle{\int{\sec^n x~dx} = \frac{\sec^{n2}x\tan x}{n1}+\frac{n2}{n1}\int{\sec^{n2}x~dx}}\) 
Inverse Trig Derivatives 

\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1t^2}} }\) 
\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = \frac{1}{\sqrt{1t^2}} }\)  
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\) 
\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = \frac{1}{1+t^2} }\)  
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 1}} }\) 
\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 1}} }\) 