The Basic Trigonometric Identities

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 2 November 2009

The Basic Trigonometric Identities

We define the exponential function by $$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\frac{x^6}{6!}+\frac{x^7}{7!}+\cdots \text{.}$$

We define the sine and cosine functions by $$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\frac{x^9}{9!}-\frac{x^{11}}{11!}+\frac{x^{13}}{13!}-\cdots \text{, and}$$ $$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}-\frac{x^{10}}{10!}+\frac{x^{12}}{12!}-\cdots \text{,}$$ respectively. We also define the tangent, cotangent, secant and cosecant functions by $$\tan x=\frac{\sin x}{\cos x}\text{,}\cot x=\frac{1}{\tan x}\text{,}\sec x=\frac{1}{\cos x}\text{,}\text{and}\csc x=\frac{1}{\sin x}\text{,}$$ respectively.

Example. Explain why $e^{ix}=\cos x+i\sin x$, if $i^2=-1$.

$$\begin{array}{rcl}{e}^{ix}& =& 1+ix+\frac{{\left(ix\right)}^2}{2!}+\frac{{\left(ix\right)}^3}{3!}+\frac{{\left(ix\right)}^4}{4!}+\frac{{\left(ix\right)}^5}{5!}+\frac{{\left(ix\right)}^6}{6!}+\frac{{\left(ix\right)}^7}{7!}+\cdots \\ & =& 1+ix+\frac{i^2{x}^2}{2!}+\frac{i^3{x}^3}{3!}+\frac{i^4{x}^4}{4!}+\frac{i^5{x}^5}{5!}+\frac{i^6{x}^6}{6!}+\frac{i^7{x}^7}{7!}+\cdots \\ & =& 1+ix+\frac{i^2{x}^2}{2!}+\frac{i·i^2{x}^3}{3!}+\frac{{\left(i^2\right)}^2{x}^4}{4!}+\frac{i·{\left(i^2\right)}^2{x}^5}{5!}+\frac{{\left(i^2\right)}^3{x}^6}{6!}+\frac{i·{\left(i^2\right)}^3{x}^7}{7!}+\cdots \\ & =& 1+ix+\frac{\left(-1\right)x^2}{2!}+\frac{i·\left(-1\right)x^3}{3!}+\frac{{\left(-1\right)}^2{x}^4}{4!}+\frac{i·{\left(-1\right)}^2{x}^5}{5!}+\frac{{\left(-1\right)}^3{x}^6}{6!}+\frac{i·{\left(-1\right)}^3{x}^7}{7!}+\cdots \\ & =& 1+ix-\frac{x^2}{2!}-i\frac{x^3}{3!}+\frac{x^4}{4!}+i\frac{x^5}{5!}-\frac{x^6}{6!}+i\frac{x^7}{7!}+\cdots \\ & =& \left(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots \right)+i\left(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots \right)\\ & =& \cos x+i\sin x\end{array}$$

Example. Expliain why $\cos \left(-x\right)=\cos x$ and $\sin \left(-x\right)=-\sin x$.

$$\begin{array}{rcl}\cos \left(-x\right)& =& 1-\frac{{\left(-x\right)}^2}{2!}+\frac{{\left(-x\right)}^4}{4!}-\frac{{\left(-x\right)}^6}{6!}+\frac{{\left(-x\right)}^8}{8!}-\frac{{\left(-x\right)}^{10}}{10!}+\frac{{\left(-x\right)}^{12}}{12!}-\cdots \\ & =& 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}-\frac{x^{10}}{10!}+\frac{x^{12}}{12!}-\cdots \\ & =& \cos x\end{array}$$ and $$\begin{array}{rcl}\sin \left(-x\right)& =& \left(-x\right)-\frac{{\left(-x\right)}^3}{3!}+\frac{{\left(-x\right)}^5}{5!}-\frac{{\left(-x\right)}^7}{7!}+\frac{{\left(-x\right)}^9}{9!}-\frac{{\left(-x\right)}^{11}}{11!}+\frac{{\left(-x\right)}^{13}}{13!}-\cdots \\ & =& -x+\frac{x^3}{3!}-\frac{x^5}{5!}+\frac{x^7}{7!}-\frac{x^9}{9!}+\frac{x^{11}}{11!}-\frac{x^{13}}{13!}+\cdots \\ & =& -\sin x\text{.}\end{array}$$

Example. Explain why ${\cos }^{2}x+{\sin }^{2}x=1$.

$$\begin{array}{rcl}1& =& e^0\\ & =& e^{ix+\left(-ix\right)}\\ & =& e^{ix}{e}^{-ix}\\ & =& e^{ix}{e}^{i\left(-x\right)}\\ & =& \left(\cos x+i\sin x\right)\left(\cos \left(-x\right)+i\sin \left(-x\right)\right)\\ & =& \left(\cos x+i\sin x\right)\left(\cos x-i\sin x\right)\\ & =& {\cos }^{2}x-i\sin x\cos x+i\sin x\cos x-i^2{\sin }^{2}x\\ & =& {\cos }^{2}x-\left(-1\right){\sin }^{2}x\\ & =& {\cos }^{2}x+{\sin }^{2}x\text{.}\end{array}$$

Example. Explain why $$\begin{array}{rcl}\cos \left(x+y\right)& =& \cos x\cos y-\sin x\sin y\text{,}\text{and}\\ \sin \left(x+y\right)& =& \sin x\cos y+\cos x\sin y\text{.}\end{array}$$

We have $$\begin{array}{rcl}\cos \left(x+y\right)+i\sin \left(x+y\right)& =& e^{i\left(x+y\right)}\\ & =& e^{ix+iy}\\ & =& e^{ix}{e}^{iy}\\ & =& \left(\cos x+i\sin x\right)\left(\cos y+i\sin y\right)\\ & =& \cos x\cos y+i\cos x\sin y+i\sin x\cos y+i^2\sin x\sin y\\ & =& \cos x\cos y+i\cos x\sin y+i\sin x\cos y-\sin x\sin y\\ & =& \left(\cos x\cos y-\sin x\sin y\right)+i\left(\cos x\sin y+\sin x\cos y\right)\text{.}\end{array}$$ Taking the real and imaginary parts of this equation gives $$\begin{array}{rcl}\cos \left(x+y\right)& =& \cos x\cos y-\sin x\sin y\text{,}\text{and}\\ \sin \left(x+y\right)& =& \sin x\cos y+\cos x\sin y\text{,}\end{array}$$ as required.

We define the hyperbolic sine and hyperbolic cosine functions by $$\sinh x=x+\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+\frac{x^9}{9!}+\frac{x^{11}}{11!}+\frac{x^{13}}{13!}+\cdots \text{, and}$$ $$\cosh x=1+\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+\frac{x^8}{8!}+\frac{x^{10}}{10!}+\frac{x^{12}}{12!}+\cdots \text{,}$$ respectively. We also define the hyperbolic tangent, hyperbolic cotangent, hyperbolic secant and hyperbolic cosecant functions by $$\tanh x=\frac{\sinh x}{\cosh x}\text{,}\coth x=\frac{1}{\tanh x}\text{,}\mathrm{sech}x=\frac{1}{\cosh x}\text{,}\text{and}\mathrm{csch}x=\frac{1}{\sinh x}\text{,}$$ respectively.

Example. Explain why $e^x=\cosh x+\sinh x$.

$$\begin{array}{rcl}{e}^x& =& 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\frac{x^6}{6!}+\frac{x^7}{7!}+\cdots \\ & =& \left(1+\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+\cdots \right)+\left(x+\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+\cdots \right)\\ & =& \cosh x+\sinh x\text{.}\end{array}$$

Example. Expliain why $\cosh \left(-x\right)=\cosh x$ and $\sinh \left(-x\right)=-\sinh x$.

$$\begin{array}{rcl}\cosh \left(-x\right)& =& 1+\frac{{\left(-x\right)}^2}{2!}+\frac{{\left(-x\right)}^4}{4!}+\frac{{\left(-x\right)}^6}{6!}+\frac{{\left(-x\right)}^8}{8!}+\frac{{\left(-x\right)}^{10}}{10!}+\frac{{\left(-x\right)}^{12}}{12!}+\cdots \\ & =& 1+\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+\frac{x^8}{8!}+\frac{x^{10}}{10!}+\frac{x^{12}}{12!}+\cdots \\ & =& \cosh x\end{array}$$ and $$\begin{array}{rcl}\sinh \left(-x\right)& =& \left(-x\right)+\frac{{\left(-x\right)}^3}{3!}+\frac{{\left(-x\right)}^5}{5!}+\frac{{\left(-x\right)}^7}{7!}+\frac{{\left(-x\right)}^9}{9!}+\frac{{\left(-x\right)}^{11}}{11!}+\frac{{\left(-x\right)}^{13}}{13!}+\cdots \\ & =& -x-\frac{x^3}{3!}-\frac{x^5}{5!}-\frac{x^7}{7!}-\frac{x^9}{9!}-\frac{x^{11}}{11!}-\frac{x^{13}}{13!}-\cdots \\ & =& -\sinh x\text{.}\end{array}$$

Example. Explain why ${\displaystyle \cosh x=\frac{e^x+e^{-x}}{2}}$ and ${\displaystyle \sinh x=\frac{e^x-e^{-x}}{2}}$.

$$\begin{array}{rcl}\frac{1}{2}\left(e^x+e^{-x}\right)& =& \frac{1}{2}\left(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\frac{x^6}{6!}+\frac{x^7}{7!}+\cdots \right.\\ & & \left.+1+\left(-x\right)+\frac{{\left(-x\right)}^2}{2!}+\frac{{\left(-x\right)}^3}{3!}+\frac{{\left(-x\right)}^4}{4!}+\frac{{\left(-x\right)}^5}{5!}+\frac{{\left(-x\right)}^6}{6!}+\frac{{\left(-x\right)}^7}{7!}+\cdots \right)\\ & =& \frac{1}{2}\left(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\frac{x^6}{6!}+\frac{x^7}{7!}+\cdots \right.\\ & & \left.+1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+\frac{x^4}{4!}-\frac{x^5}{5!}+\frac{x^6}{6!}-\frac{x^7}{7!}+\cdots \right)\\ & =& 1+\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+\frac{x^8}{8!}+\cdots \\ & =& \cosh x\end{array}$$ $$\begin{array}{rcl}\frac{1}{2}\left(e^x-e^{-x}\right)& =& \frac{1}{2}\left(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\frac{x^6}{6!}+\frac{x^7}{7!}+\cdots \right.\\ & & \left.-1-\left(-x\right)-\frac{{\left(-x\right)}^2}{2!}-\frac{{\left(-x\right)}^3}{3!}-\frac{{\left(-x\right)}^4}{4!}-\frac{{\left(-x\right)}^5}{5!}-\frac{{\left(-x\right)}^6}{6!}-\frac{{\left(-x\right)}^7}{7!}-\cdots \right)\\ & =& \frac{1}{2}\left(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\frac{x^6}{6!}+\frac{x^7}{7!}+\cdots \right.\\ & & \left.-1+x-\frac{x^2}{2!}+\frac{x^3}{3!}-\frac{x^4}{4!}+\frac{x^5}{5!}-\frac{x^6}{6!}+\frac{x^7}{7!}-\cdots \right)\\ & =& x+\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+\frac{x^9}{9!}+\cdots \\ & =& \sinh x\end{array}$$

Example. Explain why ${\cosh }^{2}x-{\sinh }^{2}x=1$.

$$\begin{array}{rcl}1& =& e^0\\ & =& e^{x+\left(-x\right)}\\ & =& e^x{e}^{-x}\\ & =& \left(\cosh x+\sinh x\right)\left(\cosh \left(-x\right)+\sinh \left(-x\right)\right)\\ & =& \left(\cosh x+\sinh x\right)\left(\cosh x-\sinh x\right)\\ & =& {\cosh }^{2}x-\sinh x\cosh x+\sinh x\cosh x-{\sinh }^{2}x\\ & =& {\cosh }^{2}x-{\sinh }^{2}x\text{.}\end{array}$$

Example. Explain why $$\begin{array}{rcl}\cosh \left(x+y\right)& =& \cosh x\cosh y+\sinh x\sinh y\text{,}\text{and}\\ \sinh \left(x+y\right)& =& \sinh x\cosh y+\cosh x\sinh y\text{.}\end{array}$$

We have $$\begin{array}{rcl}\cosh x\cosh y+\sinh x\sinh y& =& \left(\frac{e^x+e^{-x}}{2}\right)\left(\frac{e^y+e^{-y}}{2}\right)+\left(\frac{e^x-e^{-x}}{2}\right)\left(\frac{e^y-e^{-y}}{2}\right)\\ & =& \frac{e^x{e}^y+e^{-x}{e}^y+e^x{e}^{-y}+e^{-x}{e}^{-y}}{4}+\frac{e^x{e}^y-e^{-x}{e}^y-e^x{e}^{-y}+e^{-x}{e}^{-y}}{4}\\ & =& \frac{2e^x{e}^y+2e^{-x}{e}^{-y}}{4}\\ & =& \frac{e^x{e}^y+e^{-x}{e}^{-y}}{2}\\ & =& \frac{e^{x+y}+e^{-\left(x+y\right)}}{2}\\ & =& \cosh \left(x+y\right)\end{array}$$ and $$\begin{array}{rcl}\sinh x\cosh y+\cosh x\sinh y& =& \left(\frac{e^x-e^{-x}}{2}\right)\left(\frac{e^y+e^{-y}}{2}\right)+\left(\frac{e^x+e^{-x}}{2}\right)\left(\frac{e^y-e^{-y}}{2}\right)\\ & =& \frac{e^x{e}^y-e^{-x}{e}^y+e^x{e}^{-y}-e^{-x}{e}^{-y}}{4}+\frac{e^x{e}^y+e^{-x}{e}^y-e^x{e}^{-y}-e^{-x}{e}^{-y}}{4}\\ & =& \frac{2e^x{e}^y-2e^{-x}{e}^{-y}}{4}\\ & =& \frac{e^x{e}^y-e^{-x}{e}^{-y}}{2}\\ & =& \frac{e^{x+y}-e^{-\left(x+y\right)}}{2}\\ & =& \sinh \left(x+y\right)\text{.}\end{array}$$

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)