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Recipricol Identities

\sin =\frac{\text{opposite}}{\text{hypotenuse}}=\frac{1}{\csc}

\cos =\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{1}{\sec}

\tan =\frac{\text{opposite}}{\text{adjacent}}=\frac{1}{\cot}

\csc =\frac{\text{hypotenuse}}{\text{opposite}}=\frac{1}{\sin}

\sec =\frac{\text{hypotenuse}}{\text{adjacent}}=\frac{1}{\cos}

\cot =\frac{\text{adjacent}}{\text{opposite}}=\frac{1}{\tan}


Pythagorean Identities


\sin^2\left(\text{u}\right)+cos^2\left(\text{u}\right)=1

\tan^2\left(\text{u}\right)+1=sec^2\left(\text{u}\right)

1+\cot^2\left(\text{u}\right)=csc^2\left(\text{u}\right)


Even-Odd Identities

\sin\left(-\text{u}\right)=-sin\left(\text{u}\right)

\cos\left(-\text{u}\right)=-cos\left(\text{u}\right)

\tan\left(-\text{u}\right)=-tan\left(\text{u}\right)

\csc\left(-\text{u}\right)=-csc\left(\text{u}\right)

\sec\left(-\text{u}\right)=-sec\left(\text{u}\right)

\cot\left(-\text{u}\right)=-cot\left(\text{u}\right)


Co-Function Identites

\sin\left(90-\text{u}\right)=\text{cos(u)}

\cos\left(90-\text{u}\right)=\text{sin(u)}

\tan\left(90-\text{u}\right)=\text{cot(u)}

\csc\left(90-\text{u}\right)=\text{sec(u)}

\sec\left(90-\text{u}\right)=\text{csc(u)}

\cot\left(90-\text{u}\right)=\text{tan(u)}


Sum-Difference Formulas

\sin\left(\text{u}\pm \text{v}\right)=sin\left(\text{u}\right)cos\left(\text{v}\right)\pm cos\left(\text{u}\right)sin\left(\text{v}\right)

\cos\left(\text{u}\pm \text{v}\right)=cos\left(\text{u}\right)cos\left(\text{v}\right)\pm sin\left(\text{u}\right)sin\left(\text{v}\right)

\tan\left(\text{u}\pm \text{v}\right)=\frac{\text{tan(u)}\pm\text{tan(v)}}{1\pm\text{tan(u)}\cdot \text{tan(v)}}

\csc\left(\text{u}\pm \text{v}\right)=\frac{1}{sin\left(\text{u}\right)cos\left(\text{v}\right)\pm cos\left(\text{u}\right)sin\left(\text{v}\right)}

\sec\left(\text{u}\pm \text{v}\right)=\frac{1}{cos\left(\text{u}\right)cos\left(\text{v}\right)\pm sin\left(\text{u}\right)sin\left(\text{v}\right)}

\cot\left(\text{u}\pm \text{v}\right)=\frac{1\pm\text{tan(u)}\cdot \text{tan(v)}}{\text{tan(u)}\pm\text{tan(v)}}


Double Angle Formulas

\text{sin(2u)}=2\cdot \text{sin(u)}\cdot \text{cos(u)}

\text{cos(2u)}=1-2\text{sin}^2\text{(u)}

\text{tan(2u)}=\frac{2\text{tan(u)}}{1-\tan ^2\text{(u)}}

\text{csc(2u)}=\frac{1}{2\cdot \text{sin(u)}\cdot \text{cos(u)}}

\text{sec(2u)}=\frac{1}{1-2\text{sin}^2\text{(u)}}

\text{cot(2u)}=\frac{1-\tan ^2\text{(u)}}{2\text{tan(u)}}


Power Reducing Formulas

\text{sin}^2\text{(u)}=\frac{1-(1-2\cdot \text{sin}^2\text{(u)})}{2}

\text{cos}^2\text{(u)}=\frac{1+(1-2\text{sin}^2(\text{u}))}{2}

\tan ^2\text{(u)}=\frac{1+(1-2\text{sin}^2(\text{u}))}{1-(1-2\text{sin}^2(\text{u}))}

\csc ^2\text{(u)}=\frac{2}{1-(1-2\text{sin}^2(\text{u}))}

\sec ^2\text{(u)}=\frac{2}{1+(1-2\text{sin}^2(\text{u}))}

\text{cot}^2\text{(u)}=\frac{1-(1-2\text{sin}^2(\text{u}))}{1+(1-2\text{sin}^2(\text{u}))}


Half Angle Formulas

\text{sin}(\frac{\text{u}}{2})=\sqrt{\frac{1-\text{cos(u)}}{2}}

\text{cos}(\frac{\text{u}}{2})=\sqrt{\frac{1+\text{cosu}}{2}}

\text{tan}(\frac{\text{u}}{2})=\frac{1-\text{cos(u)}}{\text{sin(u)}}

\text{csc}(\frac{\text{u}}{2})=\sqrt{\frac{2}{1-\text{cos(u)}}}

\text{sec}(\frac{\text{u}}{2})=\sqrt{\frac{2}{1+\text{cos(u)}}}

\text{cot}(\frac{\text{u}}{2})=\frac{1+\text{cos(u)}}{\text{sin(u)}}


Sum to Product Formulas

\sin(\text{u})+\sin(\text{v})=2\sin(\frac{\text{u}+\text{v}}{2})\cos(\frac{\text{u}-\text{v}}{2})

\sin(\text{u})-\sin(\text{v})=2\cos(\frac{\text{u}+\text{v}}{2})\sin(\frac{\text{u}-\text{v}}{2})

\cos(\text{u})+\cos(\text{v})=2\cos(\frac{\text{u}+\text{v}}{2})\cos(\frac{\text{u}-\text{v}}{2})

\cos(\text{u})-\cos(\text{v})=-2\sin(\frac{\text{u}+\text{v}}{2})\sin(\frac{\text{u}-\text{v}}{2})

\tan(\text{u})+\tan(\text{v})=\frac{\sin\left(\text{u}+\text{v}\right)}{\cos\text{(u)}\cos\text{(v)}}

\tan(\text{u})-\tan(\text{v})=\frac{\sin\left(\text{u}-\text{v}\right)}{\cos\text{(u)}\cos\text{(v)}}


Product to Sum Formulas

\sin \left(\text{u}\right)\sin \left(\text{v}\right)=\frac{1}{2}\left(\cos \left(\text{u}-\text{v}\right)-\cos \left(\text{u}+\text{v}\right)\right)

\cos \left(\text{u}\right)\cos \left(\text{v}\right)=\frac{1}{2}\left(\cos \left(\text{u}+\text{v}\right)+\cos \left(\text{u}-\text{v}\right)\right)

\sin \left(\text{u}\right)\cos \left(\text{v}\right)=\frac{1}{2}\left(\sin \left(\text{u}+\text{v}\right)+\sin \left(\text{u}-\text{v}\right)\right)

\cos \left(\text{u}\right)\sin \left(\text{v}\right)=\frac{1}{2}\left(\sin \left(\text{u}+\text{v}\right)-\sin \left(\text{u}-\text{v}\right)\right)