## Trig and hyperbolic expressions

The exponential expression is $ex = 1+x +x22! +x33! +x44! +x55! +x66! +x77! +⋯.$

The sine and cosine expressions are $sinx = eix -e-ix 2i and cosx = eix +e-ix 2 , with i2=-1.$ The tangent, cotangent, secant and cosecant expressions are $tanx = sinxcosx, cotx = 1tanx, secx = 1cosx, and cscx = 1sinx .$

The hyperbolic sine and hyperbolic cosine expressions are $sinhx= ex- e-x 2 and coshx= ex +e-x 2 .$ The hyperbolic tangent, hyperbolic cotangent, hyperbolic secant and hyperbolic cosecant functions are $tanhx= sinhxcoshx, cothx = 1tanhx, sechx = 1coshx, and cschx= 1sinhx.$

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

 Proof. $cosx+isinx = eix +e-ix 2 +i( eix -e-ix 2i) = eix +e-ix +eix -e-ix 2 = eix.$ $\square$

Example. Prove that $sinx = x-x33! +x55! -x77! +x99! -x1111! +x1313! -⋯ and$ $cosx = 1-x22! +x44! -x66! +x88! -x1010! +x1212! -⋯ .$

 Proof. Compare coefficients of $1$ and $i$ on each side of $cosx+isinx = eix = 1+ix+ (ix) 22! + (ix) 33! + (ix) 44! + (ix) 55! + (ix) 66! + (ix) 77! +⋯ = 1+ix +i2x2 2! + i3x3 3! + i4x4 4! + i5x5 5! + i6x6 6! + i7x7 7! +⋯ = 1+ix + i2x2 2! + i· i2x3 3! + (i2)2 x4 4! + i· (i2)2 x5 5! + (i2)3 x6 6! + i· (i2)3 x77! +⋯ = 1+ix + (-1) x2 2! + i·(-1) x3 3! + (-1)2 x44! + i· (-1)2 x55! + (-1)3 x6 6! + i· (-1)3 x7 7! +⋯ = 1+ix -x22! -ix33! +x44! +ix55! -x66! -ix77! +⋯ = ( 1- x22! +x44! -x66! +⋯) +i( x- x33! +x55! -x77! +⋯) .$ □

Example. Prove that $\mathrm{cos}\left(-x\right)=\mathrm{cos}x$ and $\mathrm{sin}\left(-x\right)=-\mathrm{sin}x$.

 Proof. $cos(-x) = ei(-x) + e-i(-x) 2 = e-ix +eix 2 =cosx.$ and $sin(-x) = ei(-x) -e-i(-x) 2i = e-ix -eix 2i = -sinx.$ □

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

 Proof. $1 = e0 = e ix+ (-ix) = eix e-ix = eix ei(-x) = (cosx+isinx) (cos(-x) +isin(-x)) = (cosx+isinx) (cosx-isinx) = cos2x -isinxcosx +isinxcosx -i2sin2x = cos2x -(-1)sin2x = cos2x +sin2x.$ □

Example. Prove that $cosx+y=cosxcosy-sinxsiny,andsinx+y=sinxcosy+cosxsiny.$

 Proof. Comparing the coefficients of $1$ and $i$ on each side of $cos(x+y) +isin(x+y) = ei(x+y) = eix+iy = eix eiy = (cosx+isinx) (cosy+isiny) = cosxcosy +icosxsiny +isinxcosy +i2sinxsiny = cosxcosy +icosxsiny +isinxcosy -sinxsiny = (cosxcosy -sinxsiny) +i (cosxsiny +sinxcosy) .$ gives $cos(x+y) = cosxcosy -sinxsiny, and sin(x+y) = sinxcosy+cosxsiny .$ □

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

 Proof. $ex = 1+x +x22! +x33! +x44! +x55! +x66! +x77! +⋯ = ( 1+x22! +x44! +x66! +⋯) +( x+x33! +x55! +x77! +⋯) = coshx+sinhx .$ □

Example. Prove that $\mathrm{cosh}\left(-x\right)=\mathrm{cosh}x$ and $\mathrm{sinh}\left(-x\right)=-\mathrm{sinh}x$.

 Proof. $cosh(-x) = 1+ (-x)2 2! + (-x)4 4! + (-x)6 6! + (-x)8 8! + (-x)10 10! + (-x)12 12! +⋯ = 1+ x22! +x44! +x66! +x88! +x1010! +x1212! +⋯ = coshx$ and $sinh-x=-x+-x33!+-x55!+-x77!+-x99!+-x1111!+-x1313!+⋯=-x-x33!-x55!-x77!-x99!-x1111!-x1313!-⋯=-sinhx.$ □

Example. Prove that $sinhx=x+x33!+x55!+x77!+x99!+x1111!+x1313!+⋯ and$ $coshx=1+x22!+x44!+x66!+x88!+x1010!+x1212!+⋯.$

 Proof. $coshx = 12ex+e-x =121+x+x22!+x33!+x44!+x55!+x66!+x77!+⋯+1+-x+-x22!+-x33!+-x44!+-x55!+-x66!+-x77!+⋯=121+x+x22!+x33!+x44!+x55!+x66!+x77!+⋯+1-x+x22!-x33!+x44!-x55!+x66!-x77!+⋯=1+x22!+x44!+x66!+x88!+⋯$ $sinhx= 12ex-e-x =121+x+x22!+x33!+x44!+x55!+x66!+x77!+⋯-1--x--x22!--x33!--x44!--x55!--x66!--x77!-⋯=121+x+x22!+x33!+x44!+x55!+x66!+x77!+⋯-1+x-x22!+x33!-x44!+x55!-x66!+x77!-⋯=x+x33!+x55!+x77!+x99!+⋯$ □

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

 Proof. $1 = e0 = ex+(-x) = exe-x = (coshx+sinhx) (cosh(-x) + sinh(-x) ) = (coshx+sinhx) (coshx-sinhx) = cosh2x -sinhxcoshx +sinhxcoshx-sinh2x = cosh2x -sinh2x.$

Example. Prove that $cosh(x+y) = coshxcoshy +sinhxsinhy, and sinh(x+y) = sinhxcoshy +coshxsinhy.$

 Proof. We have $coshxcoshy+sinhxsinhy=ex+e-x2ey+e-y2+ex-e-x2ey-e-y2=exey+e-xey+exe-y+e-xe-y4+exey-e-xey-exe-y+e-xe-y4=2exey+2e-xe-y4=exey+e-xe-y2=ex+y+e-x+y2=coshx+y$ and $sinhxcoshy+coshxsinhy=ex-e-x2ey+e-y2+ex+e-x2ey-e-y2=exey-e-xey+exe-y-e-xe-y4+exey+e-xey-exe-y-e-xe-y4=2exey-2e-xe-y4=exey-e-xe-y2=ex+y-e-x+y2=sinhx+y.$ □

## 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