Triangle Identities
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Pythagoras
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Sum/Difference of angles
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cos ( x ± y ) = cos ( x ) cos ( y ) ∓ sin ( x ) sin ( y ) {\displaystyle \cos(x\pm y)=\cos(x)\cos(y)\mp \sin(x)\sin(y)}
sin ( x ± y ) = sin ( x ) cos ( y ) ± sin ( y ) cos ( x ) {\displaystyle \sin(x\pm y)=\sin(x)\cos(y)\pm \sin(y)\cos(x)}
tan ( x ± y ) = tan ( x ) ± tan ( y ) 1 ∓ tan ( x ) tan ( y ) {\displaystyle \tan(x\pm y)={\frac {\tan(x)\pm \tan(y)}{1\mp \tan(x)\tan(y)}}} Product to Sum
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2 sin ( x ) sin ( y ) = cos ( x − y ) − cos ( x + y ) {\displaystyle 2\sin(x)\sin(y)=\cos(x-y)-\cos(x+y)}
2 cos ( x ) cos ( y ) = cos ( x − y ) + cos ( x + y ) {\displaystyle 2\cos(x)\cos(y)=\cos(x-y)+\cos(x+y)}
2 sin ( x ) cos ( y ) = sin ( x − y ) + sin ( x + y ) {\displaystyle 2\sin(x)\cos(y)=\sin(x-y)+\sin(x+y)} Sum and difference to product
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A sin ( x ) + B cos ( x ) = C sin ( x + y ) {\displaystyle A\sin(x)+B\cos(x)=C\sin(x+y)} , where C = A 2 + B 2 {\displaystyle C={\sqrt {A^{2}+B^{2}}}} and y = ± arctan ( B A ) {\displaystyle y=\pm \arctan {\bigl (}{\tfrac {B}{A}}{\bigr )}}
sin ( A ) ± sin ( B ) = 2 sin ( A ± B 2 ) cos ( A ∓ B 2 ) {\displaystyle \sin(A)\pm \sin(B)=2\sin \left({\frac {A\pm B}{2}}\right)\cos \left({\frac {A\mp B}{2}}\right)}
cos ( A ) + cos ( B ) = 2 cos sin ( A + B 2 ) cos ( A − B 2 ) {\displaystyle \cos(A)+\cos(B)=2\cos \sin \left({\frac {A+B}{2}}\right)\cos \left({\frac {A-B}{2}}\right)}
cos ( A ) − cos ( B ) = − 2 sin ( A + B 2 ) sin ( A − B 2 ) {\displaystyle \cos(A)-\cos(B)=-2\sin \left({\frac {A+B}{2}}\right)\sin \left({\frac {A-B}{2}}\right)} Multiple angle
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Half angle
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Power Reduction
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sin 2 ( x ) = 1 − cos ( 2 x ) 2 {\displaystyle \sin ^{2}(x)={\frac {1-\cos(2x)}{2}}}
cos 2 ( x ) = 1 + cos ( 2 x ) 2 {\displaystyle \cos ^{2}(x)={\frac {1+\cos(2x)}{2}}}
tan 2 ( x ) = 1 − cos ( 2 x ) 1 + cos ( 2 x ) {\displaystyle \tan ^{2}(x)={\frac {1-\cos(2x)}{1+\cos(2x)}}} Even/Odd
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sin ( − x ) = − sin ( x ) {\displaystyle \sin(-x)=-\sin(x)}
cos ( − x ) = cos ( x ) {\displaystyle \cos(-x)=\cos(x)}
tan ( − x ) = − tan ( x ) {\displaystyle \tan(-x)=-\tan(x)}
csc ( − x ) = − csc ( x ) {\displaystyle \csc(-x)=-\csc(x)}
sec ( − x ) = sec ( x ) {\displaystyle \sec(-x)=\sec(x)}
cot ( − x ) = − cot ( x ) {\displaystyle \cot(-x)=-\cot(x)} Calculus
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d d x [ sin ( x ) ] = cos ( x ) {\displaystyle {\frac {d}{dx}}{\big [}\sin(x){\big ]}=\cos(x)}
d d x [ cos ( x ) ] = − sin ( x ) {\displaystyle {\frac {d}{dx}}{\big [}\cos(x){\big ]}=-\sin(x)}
d d x [ tan ( x ) ] = sec 2 ( x ) {\displaystyle {\frac {d}{dx}}{\big [}\tan(x){\big ]}=\sec ^{2}(x)}
d d x [ sec ( x ) ] = sec ( x ) tan ( x ) {\displaystyle {\frac {d}{dx}}{\big [}\sec(x){\big ]}=\sec(x)\tan(x)}
d d x [ csc ( x ) ] = − csc ( x ) cot ( x ) {\displaystyle {\frac {d}{dx}}{\big [}\csc(x){\big ]}=-\csc(x)\cot(x)}
d d x [ cot ( x ) ] = − csc 2 ( x ) {\displaystyle {\frac {d}{dx}}{\big [}\cot(x){\big ]}=-\csc ^{2}(x)}