Talk:List of trigonometric identities/removed

I have removed a lot of content from this article because, in its current form, it is too long to be manageable. Conrad.Irwin 22:16, 7 June 2007 (UTC)

Proofs

These should be at Proofs of trigonometric identities but I dont have the time right now to put them in properly, a copy and paste to there would only increase that articles problems.

Geometric proofs

These proofs apply directly only to acute angles, but the identities are still correct even when generalized to all angles. In this way, most of the trigonometric identities are deducible from elementary geometry, though many definitions and concepts have to be expanded.

sin(x + y) = sin(x) cos(y) + cos(x) sin(y)

In the figure the angle x is part of right-angled triangle ABC, and the angle y part of right-angled triangle ACD. Then construct DG perpendicular to AB and construct CE parallel to AB.

angle x = angle BAC = angle ACE = angle CDE.

EG = BC.

$\sin(x+y)\,$

$={\frac {DG}{AD}}\,$

$={\frac {EG+DE}{AD}}\,$

$={\frac {BC+DE}{AD}}\,$

$={\frac {BC}{AD}}+{\frac {DE}{AD}}\,$

$={\frac {BC}{AD}}\cdot {\frac {AC}{AC}}+{\frac {DE}{AD}}\cdot {\frac {CD}{CD}}\,$

$={\frac {BC}{AC}}\cdot {\frac {AC}{AD}}+{\frac {DE}{CD}}\cdot {\frac {CD}{AD}}\,$

$=\sin(x)\cos(y)+\cos(x)\sin(y).\,$

cos(x + y) = cos(x) cos(y) − sin(x) sin(y)

Using the above figure:

$\cos(x+y)\,$

$={\frac {AG}{AD}}\,$

$={\frac {AB-GB}{AD}}\,$

$={\frac {AB-EC}{AD}}\,$

$={\frac {AB}{AD}}-{\frac {EC}{AD}}\,$

$={\frac {AB}{AD}}\cdot {\frac {AC}{AC}}-{\frac {EC}{AD}}\cdot {\frac {CD}{CD}}\,$

$={\frac {AB}{AC}}\cdot {\frac {AC}{AD}}-{\frac {EC}{CD}}\cdot {\frac {CD}{AD}}\,$

$=\cos(x)\cos(y)-\sin(x)\sin(y).\,$

Proofs of cos(x − y) and sin(x − y) formulæ

The formulæ for cos(x − y) and sin(x − y) are easily proven using the formulæ for cos(x + y) and sin(x + y), respectively

sin(x − y) = sin(x) cos(y) − cos(x) sin(y)

To begin, we substitute y with −y into the sin(x + y) formula:

$\!\sin(x+(-y))=\sin(x)\cos(-y)+\cos(x)\sin(-y).$

Using the fact that sine is an odd function and cosine is an even function, we get

$\!\sin(x-y)=\sin(x)\cos(y)-\cos(x)\sin(y).$

cos(x − y) = cos(x) cos(y) + sin(x) sin(y)

To begin, we substitute y with −y into the cos(x + y) formula:

$\!\cos(x+(-y))=\cos(x)\cos(-y)-\sin(x)\sin(-y).$

Using the fact that sine is an odd function and cosine is an even function, we get

$\!\cos(x-y)=\cos(x)\cos(y)+\sin(x)\sin(y).$

---

Hyperbolic functions

I removed the following from the article because they are not identities of trigonometric functions. I intend to create a parallel article List of hyperbolic identities which can contain the.

$\operatorname {sinh} (\theta )=-i\sin(i\theta )={\frac {e^{\theta }-e^{-\theta }}{2}}\,$

$\operatorname {cosh} (\theta )=\cos(i\theta )={\frac {e^{\theta }+e^{-\theta }}{2}}\,$

$\operatorname {tanh} (\theta )={\frac {\operatorname {sinh} (\theta )}{\operatorname {cosh} (\theta )}}={\frac {e^{\theta }-e^{-\theta }}{e^{\theta }+e^{-\theta }}}={\frac {e^{2\theta }-1}{e^{2\theta }+1}}\,$

$\operatorname {coth} (\theta )={\frac {\operatorname {cosh} (\theta )}{\operatorname {sinh} (\theta )}}={\frac {e^{\theta }+e^{-\theta }}{e^{\theta }-e^{-\theta }}}={\frac {e^{2\theta }+1}{e^{2\theta }-1}}\,$

$\operatorname {sech} (\theta )={\frac {1}{\operatorname {cosh} (\theta )}}=\operatorname {sec} (i\theta )={\frac {2}{e^{\theta }+e^{-\theta }}}\,$

$\operatorname {csch} (\theta )={\frac {1}{\operatorname {sinh} (\theta )}}=i\cos(i\theta )={\frac {2}{e^{\theta }-e^{-\theta }}}\,$

$\operatorname {versinh} (\theta )=1-\operatorname {cosh} (\theta )=1-\cos(i\theta )=1-{\frac {e^{\theta }+e^{-\theta }}{2}}\,$

$\operatorname {vercosh} (\theta )=1-\operatorname {sinh} (\theta )=1+i\sin(i\theta )=1-{\frac {e^{\theta }-e^{-\theta }}{2}}\,$

$\operatorname {exsech} (\theta )=\operatorname {sech} (\theta )-1={\frac {1}{\operatorname {cosh} (\theta )}}-1={\frac {2}{e^{\theta }+e^{-\theta }}}-1\,$

$\operatorname {excsch} (\theta )=\operatorname {csch} (\theta )-1={\frac {1}{\operatorname {sinh} (\theta )}}-1={\frac {2}{e^{\theta }-e^{-\theta }}}-1\,$

$\operatorname {arcsinh} (\theta )=\ln(\theta +{\sqrt {\theta ^{2}+1}})\,$

$\operatorname {arccosh} (\theta )=\ln(\theta +{\sqrt {\theta ^{2}-1}})\,$

$\operatorname {arctanh} (\theta )={\frac {\ln({\frac {i+\theta }{i-\theta }})}{2}}\,$

$\operatorname {arccoth} (\theta )=\operatorname {arctanh} (-\theta )={\frac {\ln({\frac {i-\theta }{i+\theta }})}{2}}\,$

$\operatorname {arcsech} (\theta )=\operatorname {arccosh} (\theta ^{-1})=\ln(\theta ^{-1}+{\sqrt {\theta ^{-2}-1}})\,$

$\operatorname {arccsch} (\theta )=\operatorname {arcsinh} (\theta ^{-1})=\ln(\theta ^{-1}+{\sqrt {\theta ^{-2}+1}})\,$

$\operatorname {arcversinh} (\theta )={\frac {\operatorname {arccos} (1-\theta )}{i}}\,$

$\operatorname {arcvercosh} (\theta )={\frac {\operatorname {arcsin} {\frac {\theta -1}{i}}}{i}}\,$

$\operatorname {arcexsech} (\theta )=\operatorname {arcsech} (\theta +1)=\operatorname {arccosh} ((\theta +1)^{-1})=\ln((\theta +1)^{-1}+{\sqrt {(\theta +1)^{-2}-1}})\,$

$\operatorname {arcexcsch} (\theta )=\operatorname {arccsch} (\theta +1)=\operatorname {arcsinh} ((\theta +1)^{-1})=\ln((\theta +1)^{-1}+{\sqrt {(\theta +1)^{-2}+1}})\,$

$\operatorname {cish} (\theta )=\operatorname {cosh} (\theta )+i\ \operatorname {sinh} (\theta )={\frac {e^{\theta }+e^{-\theta }}{2}}+i{\frac {e^{\theta }-e^{-\theta }}{2}}=\operatorname {cos} (i\theta )+\operatorname {sin} (i\theta )\,$

$\operatorname {arccish} (\theta )={\frac {\operatorname {arcsin} (\theta ^{2}-1)}{2i}}={\frac {-\ln(i(\theta ^{2}-1)+{\sqrt {1-(\theta ^{2}-1)^{2}}})}{2}}\,$

---

Exponential Definitions of historical functions

I removed the following from the article because they are trivial to calculate from the information given above, and I believe that anyone sufficiently interested in these functions could work them out by themselves.

$\operatorname {versin} (\theta )=1-\cos(\theta )=1-{\frac {e^{i\theta }+e^{-i\theta }}{2}}\,$

$\operatorname {vercos} (\theta )=1-\sin(\theta )=1-{\frac {e^{i\theta }-e^{-i\theta }}{2i}}\,$

$\operatorname {exsec} (\theta )=\operatorname {sec} (\theta )-1\ ={\frac {1}{\cos(\theta )}}-1={\frac {1}{({\frac {e^{i\theta }+e^{-i\theta }}{2}})}}-1={\frac {2}{e^{i\theta }+e^{-i\theta }}}-1\,$

$\operatorname {excsc} (\theta )=\operatorname {csc} (\theta )-1\ ={\frac {1}{\sin(\theta )}}-1={\frac {1}{({\frac {e^{i\theta }-e^{-i\theta }}{2i}})}}-1={\frac {2i}{e^{i\theta }-e^{-i\theta }}}-1\,$

$\operatorname {arcversin} (\theta )=\arccos(1-\theta )=-i\ln(1-\theta +i{\sqrt {1-(1-\theta )^{2}}})\,$

$\operatorname {arcvercos} (\theta )=\operatorname {arcsin} (1-\theta )=-i\ln(i-i\theta +{\sqrt {1-(1-\theta )^{2}}})\,$

$\operatorname {arcexsec} (\theta )=\operatorname {arcsec}(1+\theta )=-i\ln((\theta +1)^{-1}+i{\sqrt {1-(1+\theta )^{2}}})\,$

$\operatorname {arcexcsc} (\theta )=\operatorname {arccsc}(1+\theta )=-i\ln(i(\theta +1)^{-1}+{\sqrt {1-(1+\theta )^{2}}})\,$

---

Derivations

I removed the following from the article because I dont see that this is appropriate content for an enclyclopedia, it might be found in a maths textbook, in which case the following cold be used in a wikibook.

Tan derivation

$\operatorname {tan} (\theta )\,$$={\frac {\operatorname {sin} (\theta )}{\operatorname {cos} (\theta )}}\,$$={\frac {\frac {e^{i\theta }-e^{-i\theta }}{2i}}{\frac {e^{i\theta }+e^{-i\theta }}{2}}}\,$$={\frac {2(e^{i\theta }-e^{-i\theta })}{2i(e^{i\theta }+e^{-i\theta })}}\,$$={\frac {e^{i\theta }-e^{-i\theta }}{i(e^{i\theta }+e^{-i\theta })}}\,$

Cot derivation

$\cot(\theta )={\frac {\cos(\theta )}{\sin(\theta )}}={\frac {1}{\tan \theta }}={\frac {\left({\frac {e^{i\theta }+e^{-i\theta }}{2}}\right)}{\left({\frac {e^{i\theta }-e^{-i\theta }}{2i}}\right)}}={\frac {2i(e^{i\theta }+e^{-i\theta })}{2(e^{i\theta }-e^{-i\theta })}}={\frac {i(e^{i\theta }+e^{-i\theta })}{e^{i\theta }-e^{-i\theta }}}\,$

Sec derivation

${\begin{aligned}\operatorname {sec} (\theta )&{}={\frac {1}{\operatorname {cos} (\theta )}}\\\\&{}={\frac {1}{({\frac {e^{i\theta }+e^{-i\theta }}{2}})}}\\\\&{}={\frac {2}{e^{i\theta }+e^{-i\theta }}}\end{aligned}}$

Csc derivation

${\begin{aligned}\operatorname {csc} (\theta )&{}={\frac {1}{\operatorname {sin} (\theta )}}\\\\&{}={\frac {1}{({\frac {e^{i\theta }-e^{-i\theta }}{2i}})}}\\\\&{}={\frac {2i}{e^{i\theta }-e^{-i\theta }}}\end{aligned}}$

Cis derivation

$\operatorname {cis} (x)\,$

$=\operatorname {cos} (x)+i\ \operatorname {sin} (x)\,$

$={\frac {e^{ix}+e^{-ix}}{2}}+i{\frac {e^{ix}-e^{-ix}}{2i}}\,$

$={\frac {e^{ix}+e^{-ix}}{2}}+{\frac {e^{ix}-e^{-ix}}{2}}\,$

$={\frac {(e^{ix}+e^{-ix})+(e^{ix}-e^{-ix})}{2}}\,$

$={\frac {e^{ix}+e^{-ix}+e^{ix}-e^{-ix}}{2}}\,$

$={\frac {e^{ix}+e^{ix}+e^{-ix}-e^{-ix}}{2}}\,$

$={\frac {e^{ix}+e^{ix}+0}{2}}\,$

$={\frac {e^{ix}+e^{ix}}{2}}\,$

$={\frac {2(e^{ix})}{2}}$

$=e^{ix}\,$

Tanh derivation

${\begin{aligned}\operatorname {tanh} (x)&{}={\frac {\operatorname {sinh} (x)}{\operatorname {cosh} (x)}}\\\\&{}={\dfrac {\left({\dfrac {e^{x}-e^{-x}}{2}}\right)}{\left({\dfrac {e^{x}+e^{-x}}{2}}\right)}}\\\\&{}={\dfrac {\left({\dfrac {e^{x}-e^{-x}}{2}}\right)}{e^{x}+e^{-x}}}2\\\\&{}={\dfrac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}\\\\&{}={\dfrac {\left({\dfrac {e^{x}-e^{-x}}{e^{-x}}}\right)}{\left({\dfrac {e^{x}+e^{-x}}{e^{-x}}}\right)}}\\\\&{}={\dfrac {{\dfrac {e^{x}}{e^{-x}}}-1}{{\dfrac {e^{x}}{e^{-x}}}+1}}\\\\&{}={\dfrac {e^{x}e^{x}-1}{e^{x}e^{x}+1}}\\\\&{}={\dfrac {(e^{x})^{2}-1}{(e^{x})^{2}+1}}\\\\&{}={\frac {e^{2x}-1}{e^{2x}+1}}.\end{aligned}}$

Coth derivation

${\begin{aligned}\coth(x)&{}={\frac {\cosh(x)}{\sinh(x)}}\\\\&{}={\frac {({\frac {e^{x}+e^{-x}}{2}})}{({\frac {e^{x}-e^{-x}}{2}})}}\\\\&{}={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}\\\\&{}={\frac {({\frac {e^{x}+e^{-x}}{e^{-x}}})}{({\frac {e^{x}-e^{-x}}{e^{-x}}})}}\\\\&{}={\frac {{\frac {e^{x}}{e^{-x}}}+1}{{\frac {e^{x}}{e^{-x}}}-1}}\\\\&{}={\frac {e^{x}e^{x}+1}{e^{x}e^{x}-1}}\\\\&{}={\frac {(e^{x})^{2}+1}{(e^{x})^{2}-1}}\\\\&{}={\frac {e^{2x}+1}{e^{2x}-1}}\end{aligned}}$

Sech derivation

${\begin{aligned}\operatorname {sech} (x)&{}={\frac {1}{\cosh(x)}}\\\\&{}={\frac {1}{\left({\dfrac {e^{x}+e^{-x}}{2}}\right)}}\\\\&{}={\frac {2}{e^{x}+e^{-x}}}\end{aligned}}$

Sech alternative derivation

${\begin{aligned}\operatorname {sech} (x)&{}={\frac {1}{\operatorname {cosh} (x)}}\\&{}={\frac {1}{\operatorname {cos} (ix)}}\\&{}={\frac {1}{({\frac {e^{i^{2}x}+e^{-i^{2}x}}{2}})}}\\&{}={\frac {2}{e^{i^{2}x}+e^{-i^{2}x}}}\\&{}={\frac {2}{e^{-x}+e^{-(-1)x}}}\\&{}={\frac {2}{e^{-x}+e^{x}}}\\&{}={\frac {2}{e^{x}+e^{-x}}}\end{aligned}}$

Csch derivation

${\begin{aligned}\operatorname {csch} (x)&{}={\frac {1}{\operatorname {sinh} (x)}}\\&{}={\frac {1}{({\frac {e^{x}-e^{-x}}{2}})}}\\&{}={\frac {1}{e^{x}-e^{-x}}}2\\&{}={\frac {2}{e^{x}-e^{-x}}}\end{aligned}}$

Csch alternative derivation

$\operatorname {csch} (x)=\,$

$={\frac {1}{\operatorname {sinh} (x)}}=\,$

$={\frac {1}{-i\ \operatorname {sin} (ix)}}=\,$

$={\frac {1}{-i{\frac {e^{i^{2}x}-e^{-i^{2}x}}{2i}}}}=\,$

$={\frac {1}{-i{\frac {e^{-x}-e^{x}}{2i}}}}=\,$

$={\frac {1}{-i{\frac {-e^{x}+e^{-x}}{2i}}}}=\,$

$={\frac {1}{i{\frac {e^{x}-e^{-x}}{2i}}}}=\,$

$={\frac {1}{\frac {e^{x}-e^{-x}}{2}}}=\,$

$={\frac {1}{e^{x}-e^{-x}}}2=\,$

$={\frac {2}{e^{x}-e^{-x}}}\,$

Arcsec derivation

$\operatorname {sec} (x)=\theta \,$

${\frac {1}{\operatorname {cos} (x)}}=\theta \,$

$\operatorname {cos} (x)=\theta ^{-1}\,$

$x=\operatorname {arccos} (\theta ^{-1})\,$

Arccsc derivation

$\operatorname {csc} (x)=\theta \,$

${\frac {1}{\operatorname {sin} (x)}}=\theta \,$

$\operatorname {sin} (x)=\theta ^{-1}\,$

$x=\operatorname {arcsin} (\theta ^{-1})\,$

Arcversin derivation

$\operatorname {versin} (x)=\theta \,$

$1-\operatorname {cos} (x)=\theta \,$

$\operatorname {cos} (x)=1-\theta \,$

$x=\operatorname {arccos} (1-\theta )\,$

Arcexsec derivation

$\operatorname {exsec} (x)=\theta \,$

$\operatorname {sec} (x)-1=\theta \,$

$\operatorname {sec} (x)=\theta +1\,$

$x=\operatorname {arcsec} (\theta +1)\,$

Arcexcsc derivation

$\operatorname {excsc} (x)=\theta \,$

$\operatorname {csc} (x)-1=\theta \,$

$\operatorname {csc} (x)=\theta +1\,$

$x=\operatorname {arccsc} (\theta +1)\,$

Arcsech derivation

$\theta =\operatorname {sech} (x)\,$

$\theta ={\frac {1}{\operatorname {cosh} (x)}}\,$

${\frac {1}{\theta }}=\operatorname {cosh} (x)\,$

$\theta ^{-1}=\operatorname {cosh} (x)\,$

$\operatorname {arccosh} (\theta ^{-1})=x\,$

Arccsch derivation

$\theta =\operatorname {csch} (x)\,$

$\theta ={\frac {1}{\operatorname {sinh} (x)}}\,$

${\frac {1}{\theta }}=\operatorname {sinh} (x)\,$

$\theta ^{-1}=\operatorname {sinh} (x)\,$

$\operatorname {arcsinh} (\theta ^{-1})=x\,$

Arcversinh derivation

$\theta =\operatorname {versinh} (x)\,$

$\theta =1-\operatorname {cosh} (x)\,$

$\theta =1-\operatorname {cos} (ix)\,$

$1-\theta =\operatorname {cos} (ix)\,$

$\operatorname {arccos} (1-\theta )=ix\,$

$x={\frac {\operatorname {arccos} (1-\theta )}{i}}\,$

Arcexsech derivation

$\theta =\operatorname {exsech} (x)\,$

$\theta =\operatorname {sech} (x)-1\,$

$\theta +1=\operatorname {sech} (x)\,$

$x=\operatorname {arcsech} (\theta +1)\,$

Arcexcsch derivation

$\theta =\operatorname {excsch} (x)\,$

$\theta =\operatorname {csch} (x)-1\,$

$\theta +1=\operatorname {csch} (x)\,$

$x=\operatorname {arccsch} (\theta +1)\,$

Arccis derivation

$\theta =\operatorname {cis} (x)\,$

$\theta =e^{ix}\,$

$\ln \theta =\ln e^{ix}\,$

$\ln \theta =ix\,$

${\frac {\ln \theta }{i}}={\frac {ix}{i}}\,$

${\frac {\ln \theta }{i}}=x\,$

Arccish derivation

$\theta =\operatorname {cish} (x)\,$

$\theta =\operatorname {cosh} (x)+i\ \operatorname {sinh} (x)\,$

$\theta ^{2}=(\operatorname {cosh} (x)+i\ \operatorname {sinh} (x))^{2}\,$

$\theta ^{2}=\operatorname {cosh^{2}} (x)+2i\ \operatorname {cosh} (x)\operatorname {sinh} (x)-\operatorname {sinh^{2}} (x)\,$

$\theta ^{2}=\operatorname {cosh^{2}} (x)+2i\ \operatorname {cos} (ix)\operatorname {sinh} (x)-\operatorname {sinh^{2}} (x)\,$

$\theta ^{2}=\operatorname {cosh^{2}} (x)+2\ \operatorname {cos} (ix)\operatorname {sin} (ix)-\operatorname {sinh^{2}} (x)\,$

$\theta ^{2}=\operatorname {cosh^{2}} (x)+\operatorname {sin} (ix+ix)-\operatorname {sin} (ix-ix)-\operatorname {sinh^{2}} (x)\,$

$\theta ^{2}=\operatorname {cosh^{2}} (x)+\operatorname {sin} (2ix)-\operatorname {sin} (0)-\operatorname {sinh^{2}} (x)\,$

$\theta ^{2}=\operatorname {cosh^{2}} (x)+\operatorname {sin} (2ix)-0-\operatorname {sinh^{2}} (x)\,$

$\theta ^{2}=\operatorname {cosh^{2}} (x)+\operatorname {sin} (2ix)-\operatorname {sinh^{2}} (x)\,$

$\theta ^{2}=\operatorname {cosh^{2}} (x)-\operatorname {sinh^{2}} (x)+\operatorname {sin} (2ix)\,$

$\theta ^{2}=1+\operatorname {sin} (2ix)\,$

$\theta ^{2}-1=\operatorname {sin} (2ix)\,$

$\operatorname {arcsin} (\theta ^{2}-1)=2ix\,$

$x={\frac {\operatorname {arcsin} (\theta ^{2}-1)}{2i}}\,$