List of Identities

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Reciprocal Identities

\[ \begin{align*} {\color{flatlightblue}\csc(\theta)} &= \dfrac{1}{\color{flatblue}\sin(\theta)} & {\color{flatlightred}\sec(\theta)} &= \dfrac{1}{\color{flatred}\cos(\theta)} & {\color{flatlightgreen}\cot(\theta)} &= \dfrac{1}{\color{flatgreen}\tan(\theta)} \\ {\color{flatblue}\sin(\theta)} &= \dfrac{1}{\color{flatlightblue}\csc(\theta)} & {\color{flatred}\cos(\theta)} &= \dfrac{1}{\color{flatlightred}\sec(\theta)} & {\color{flatgreen}\tan(\theta)} &= \dfrac{1}{\color{flatlightgreen}\cot(\theta)} \end{align*} \]

Quotient Identities

\[ \begin{align*} {\color{flatgreen}\tan(\theta)} &= \dfrac{\color{flatblue}\sin(\theta)}{\color{flatred}\cos(\theta)} & {\color{flatlightgreen}\cot(\theta)} &= \dfrac{\color{flatred}\cos(\theta)}{\color{flatblue}\sin(\theta)} & \end{align*} \]

Pythagorean Identities

\[ \begin{align*} {\color{flatblue}\sin^2(\theta)}+{\color{flatred}\cos^2(\theta)} &= {1} \\ {\color{flatgreen}\tan^2(\theta)}\ \ \ +\ \ \ {1}\ \ \ &= {\color{flatlightred}\sec^2(\theta)} \\ {1}\ \ \ +\ \ \ {\color{flatlightgreen}\cot^2(\theta)} &= {\color{flatlightblue}\csc^2(\theta)} \end{align*} \]

Cofunction Identities

\[ \begin{align*} {\sin({\color{flatpurple}\theta})} &= {\cos({\color{flatpink}\tfrac\pi2-\theta})} & {\tan({\color{flatpurple}\theta})} &= {\cot({\color{flatpink}\tfrac\pi2-\theta})} & {\sec({\color{flatpurple}\theta})} &= {\csc({\color{flatpink}\tfrac\pi2-\theta})} \\ {\cos({\color{flatpurple}\theta})} &= {\sin({\color{flatpink}\tfrac\pi2-\theta})} & {\cot({\color{flatpurple}\theta})} &= {\tan({\color{flatpink}\tfrac\pi2-\theta})} & {\csc({\color{flatpurple}\theta})} &= {\sec({\color{flatpink}\tfrac\pi2-\theta})} \end{align*} \]

Even-Odd Identities

\[ \begin{align*} {\color{flatblue}\sin({\color{black}-}\theta)} &= -{\color{flatblue}\sin(\theta)} & {\color{flatred}\cos({\color{black}-}\theta)} &= {\color{flatred}\cos(\theta)} \\ {\color{flatlightblue}\csc({\color{black}-}\theta)} &= -{\color{flatlightblue}\csc(\theta)} & {\color{flatlightred}\sec({\color{black}-}\theta)} &= {\color{flatlightred}\sec(\theta)} \\ {\color{flatgreen}\tan({\color{black}-}\theta)} &= -{\color{flatgreen}\tan(\theta)} \\ {\color{flatlightgreen}\cot({\color{black}-}\theta)} &= -{\color{flatlightgreen}\cot(\theta)} \\ \end{align*} \]

Periodic Identities

\[ \begin{align*} {\color{flatblue}\sin(\theta{\color{black}\ +\ 2\pi})} &= {\color{flatblue}\sin(\theta)} & {\color{flatgreen}\tan(\theta{\color{black}\ +\ \pi})} &= {\color{flatgreen}\tan(\theta)} \\ {\color{flatlightblue}\csc(\theta{\color{black}\ +\ 2\pi})} &= {\color{flatlightblue}\csc(\theta)} & {\color{flatlightgreen}\cot(\theta{\color{black}\ +\ \pi})} &= {\color{flatlightgreen}\cot(\theta)} \\ {\color{flatred}\cos(\theta{\color{black}\ +\ 2\pi})} &= {\color{flatred}\cos(\theta)} \\ {\color{flatlightred}\sec(\theta{\color{black}\ +\ 2\pi})} &= {\color{flatlightred}\sec(\theta)} \end{align*} \]

Sum and Difference Identities

\[ \begin{align*} \sin(\alpha+\beta) &= \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)\\ \sin(\alpha-\beta) &= \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta)\\[3 ex] \cos(\alpha+\beta) &= \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)\\ \cos(\alpha-\beta) &= \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)\\[3 ex] \tan(\alpha+\beta) &= \dfrac{\tan(\alpha)+\tan(\beta)}{1-\tan(\alpha)\tan(\beta)}\\ \tan(\alpha-\beta) &= \dfrac{\tan(\alpha)-\tan(\beta)}{1+\tan(\alpha)\tan(\beta)} \end{align*} \]

Double Angle Identities

\[ \begin{align*} \sin(2\theta) &= 2\sin(\theta)\cos(\theta)\\[3 ex] \cos(2\theta) &= \cos^2(\theta)-\sin^2(\theta)\\ &= 1-2\sin^2(\theta)\\ &= 2\cos^2(\theta)-1\\[3 ex] \tan(2\theta) &= \dfrac{2\tan(\theta)}{1-\tan^2(\theta)} \end{align*} \]

Half Angle Identities

\[ \begin{align*} \left|\sin\left(\dfrac\theta2\right)\right| &= \sqrt{\dfrac{1-\cos(\theta)}2}\\[3 ex] \left|\cos\left(\dfrac\theta2\right)\right| &= \sqrt{\dfrac{1+\cos(\theta)}2}\\[3 ex] \tan\left(\dfrac\theta2\right) &= \dfrac{1-\cos(\theta)}{\sin(\theta)}\\ &= \dfrac{\sin(\theta)}{1+\cos(\theta)} \end{align*} \]

This is all the identities we'll use in class. There are a few others that sometimes come in handy for other applications, so I'm including them below for the sake of completeness. It's a good exercise to try to prove them from the sum and difference identities, using systems of equations!

Product-to-Sum Identities

\[ \begin{align*} \sin(\alpha)\sin(\beta) &= \tfrac12\big(\cos(\alpha-\beta)-\cos(\alpha+\beta)\big)\\ \cos(\alpha)\cos(\beta) &= \tfrac12\big(\cos(\alpha-\beta)+\cos(\alpha+\beta)\big)\\ \sin(\alpha)\cos(\beta) &= \tfrac12\big(\sin(\alpha+\beta)+\sin(\alpha-\beta)\big)\\ \cos(\alpha)\sin(\beta) &= \tfrac12\big(\sin(\alpha+\beta)-\sin(\alpha-\beta)\big) \end{align*} \]

Sum-to-Product Identities

\[ \begin{align*} \sin(\alpha)+\sin(\beta) &= 2\sin\left(\dfrac{\alpha+\beta}{2}\right)\cos\left(\dfrac{\alpha-\beta}{2}\right)\\ \sin(\alpha)-\sin(\beta) &= 2\cos\left(\dfrac{\alpha+\beta}{2}\right)\sin\left(\dfrac{\alpha-\beta}{2}\right)\\ \cos(\alpha)+\cos(\beta) &= 2\cos\left(\dfrac{\alpha+\beta}{2}\right)\cos\left(\dfrac{\alpha-\beta}{2}\right)\\ \cos(\alpha)-\cos(\beta) &= -2\sin\left(\dfrac{\alpha+\beta}{2}\right)\sin\left(\dfrac{\alpha-\beta}{2}\right) \end{align*} \]

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Posted by Bill Shillito