Trig and hyperbolic expressions

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

Last updates: 17 June 2011

Trig and hyperbolic expressions

The exponential expression is $$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\frac{x^6}{6!}+\frac{x^7}{7!}+\cdots \text{.}$$

The sine and cosine expressions are $$\sin x=\frac{e^{ix}-e^{-ix}}{2i}\text{and}\cos x=\frac{e^{ix}+e^{-ix}}{2},\text{with}{i}^2=-1.$$ The tangent, cotangent, secant and cosecant expressions are $$\tan x=\frac{\sin x}{\cos x}\text{,}\cot x=\frac{1}{\tan x}\text{,}\sec x=\frac{1}{\cos x}\text{,}\text{and}\csc x=\frac{1}{\sin x}\text{.}$$

The hyperbolic sine and hyperbolic cosine expressions are $$\sinh x=\frac{e^x-e^{-x}}{2}\text{and}\cosh x=\frac{e^x+e^{-x}}{2}.$$ The hyperbolic tangent, hyperbolic cotangent, hyperbolic secant and hyperbolic cosecant functions are $$\tanh x=\frac{\sinh x}{\cosh x}\text{,}\coth x=\frac{1}{\tanh x}\text{,}\mathrm{sech}x=\frac{1}{\cosh x}\text{,}\text{and}\mathrm{csch}x=\frac{1}{\sinh x}\text{.}$$

Example. Prove that $e^{ix}=\cos x+i\sin x$.

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

Example. Prove that $$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\frac{x^9}{9!}-\frac{x^{11}}{11!}+\frac{x^{13}}{13!}-\cdots \text{and}$$ $$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}-\frac{x^{10}}{10!}+\frac{x^{12}}{12!}-\cdots \hspace{0.17em}\text{.}$$

		Proof.
	Compare coefficients of $1$ and $i$ on each side of $$\begin{array}{rcl}\cos x+i\sin x& =& e^{ix}\\ & =& 1+ix+\frac{{\left(ix\right)}^2}{2!}+\frac{{\left(ix\right)}^3}{3!}+\frac{{\left(ix\right)}^4}{4!}+\frac{{\left(ix\right)}^5}{5!}+\frac{{\left(ix\right)}^6}{6!}+\frac{{\left(ix\right)}^7}{7!}+\cdots \\ & =& 1+ix+\frac{i^2{x}^2}{2!}+\frac{i^3{x}^3}{3!}+\frac{i^4{x}^4}{4!}+\frac{i^5{x}^5}{5!}+\frac{i^6{x}^6}{6!}+\frac{i^7{x}^7}{7!}+\cdots \\ & =& 1+ix+\frac{i^2{x}^2}{2!}+\frac{i·i^2{x}^3}{3!}+\frac{{\left(i^2\right)}^2{x}^4}{4!}+\frac{i·{\left(i^2\right)}^2{x}^5}{5!}+\frac{{\left(i^2\right)}^3{x}^6}{6!}+\frac{i·{\left(i^2\right)}^3{x}^7}{7!}+\cdots \\ & =& 1+ix+\frac{(-1)x^2}{2!}+\frac{i·(-1)x^3}{3!}+\frac{{(-1)}^2{x}^4}{4!}+\frac{i·{(-1)}^2{x}^5}{5!}+\frac{{(-1)}^3{x}^6}{6!}+\frac{i·{(-1)}^3{x}^7}{7!}+\cdots \\ & =& 1+ix-\frac{x^2}{2!}-i\frac{x^3}{3!}+\frac{x^4}{4!}+i\frac{x^5}{5!}-\frac{x^6}{6!}-i\frac{x^7}{7!}+\cdots \\ & =& (1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots )+i(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots ).\end{array}$$ □

Example. Prove that $\cos (-x)=\cos x$ and $\sin (-x)=-\sin x$.

		Proof.
	$$\begin{array}{rcl}\cos (-x)& =& \frac{e^{i(-x)}+e^{-i(-x)}}{2}\\ & =& \frac{e^{-ix}+e^{ix}}{2}\\ & =& \cos x.\end{array}$$ and $$\begin{array}{rcl}\sin (-x)& =& \frac{e^{i(-x)}-e^{-i(-x)}}{2i}\\ & =& \frac{e^{-ix}-e^{ix}}{2i}\\ & =& -\sin x.\end{array}$$ □

Example. Prove that ${\cos }^{2}x+{\sin }^{2}x=1$.

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

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

		Proof.
	Comparing the coefficients of $1$ and $i$ on each side of $$\begin{array}{rcl}\cos (x+y)+i\sin (x+y)& =& e^{i(x+y)}\\ & =& e^{ix+iy}\\ & =& e^{ix}{e}^{iy}\\ & =& (\cos x+i\sin x)(\cos y+i\sin y)\\ & =& \cos x\cos y+i\cos x\sin y+i\sin x\cos y+i^2\sin x\sin y\\ & =& \cos x\cos y+i\cos x\sin y+i\sin x\cos y-\sin x\sin y\\ & =& (\cos x\cos y-\sin x\sin y)+i(\cos x\sin y+\sin x\cos y)\text{.}\end{array}$$ gives $$\begin{array}{rcl}\cos (x+y)& =& \cos x\cos y-\sin x\sin y,\text{and}\\ \sin (x+y)& =& \sin x\cos y+\cos x\sin y\text{.}\end{array}$$ □

Example. Prove that $e^x=\cosh x+\sinh x$.

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

Example. Prove that $\cosh (-x)=\cosh x$ and $\sinh (-x)=-\sinh x$.

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

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

Example. Prove that ${\cosh }^{2}x-{\sinh }^{2}x=1$.

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

Example. Prove that $$\begin{array}{rcl}\cosh (x+y)& =& \cosh x\cosh y+\sinh x\sinh y\text{,}\text{and}\\ \sinh (x+y)& =& \sinh x\cosh y+\cosh x\sinh y\text{.}\end{array}$$

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

Notes and References

The student should learn to do these proofs at the same time that the trig expressions are introduced.

References

[Bou] N. Bourbaki, Algebra II, Chapters 4–7 Translated from the 1981 French edition by P. M. Cohn and J. Howie, Reprint of the 1990 English edition, Springer-Verlag, Berlin, 2003. viii+461 pp. ISBN: 3-540-00706-7. MR1994218

[Mac] I.G. Macdonald, Symmetric functions and Hall polynomials, Second edition, Oxford University Press, 1995. ISBN: 0-19-853489-2 MR1354144

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