$$\begin{equation} a^m \times b^n = a^{m+n} \end{equation}$$ $$\begin{equation} a^m \div b^n = a^{m-n} \end{equation}$$ $$\begin{equation} (a^m)^n = a^{mn} \end{equation}$$ $$\begin{equation} (ab)^n = a^n b^n \end{equation}$$ $$\begin{equation} \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \end{equation}$$ $$\begin{equation} a^0 = 1 \end{equation}$$ $$\begin{equation} a^{-n} = \frac{1}{a^n} \end{equation}$$ $$\begin{equation} a^{\frac{m}{n}} = \sqrt[n]{a^m} \end{equation}$$

$$\begin{equation} \sin x=\cos(90^\circ-x) \end{equation}$$ $$\begin{equation} \cos x=\sin(90^\circ-x) \end{equation}$$ $$\begin{equation} \sin x=-\sin(-x) \end{equation}$$ $$\begin{equation} \cos x=\cos(-x) \end{equation}$$ $$\begin{equation} \sin x=\cos(x-90^\circ) \end{equation}$$ $$\begin{equation} \tan x=\frac{\sin x}{\cos x} \end{equation}$$

$$\begin{equation} \sin^2 x + \cos^2 x = 1\end{equation}$$

$$\begin{equation} \sin(x\pm y)=\sin x \cos y \pm \cos x sin y\end{equation}$$ $$\begin{equation} \cos(x\pm y)=\cos x \cos y \mp \sin x \sin y\end{equation}$$ $$\begin{equation} \sin 2x =2\sin x \cos x \end{equation}$$ $$\begin{equation} \cos 2x =\cos^2 x -\sin^2 x = 2\cos^2 x -1 = 1 -2\sin^2 x\end{equation}$$ $$\begin{equation} \tan(x\pm y)=\frac{\tan \pm \tan y}{1 \mp \tan x\tan y } \end{equation}$$

$$\begin{equation} \cos x \cos y =\frac{1}{2} \left[\cos(x-y)+\cos(x+y) \right] \end{equation}$$ $$\begin{equation} \sin x \sin y =\frac{1}{2} \left[\cos(x-y)-\cos(x+y) \right] \end{equation}$$ $$\begin{equation} \sin x \cos y =\frac{1}{2} \left[\sin(x+y)+\sin(x-y) \right] \end{equation}$$

$$\begin{equation} \sin x \pm \sin y =2 \sin\left(\frac{x \pm y}{2}\right) \cos\left(\frac{y \mp x}{2}\right) \end{equation}$$ $$\begin{equation} \cos x + \cos y =2 \cos\left(\frac{x + y}{2}\right) \cos\left(\frac{y - x}{2}\right) \end{equation}$$ $$\begin{equation} \cos x - \cos y =-2 \sin\left(\frac{x + y}{2}\right) \sin\left(\frac{y - x}{2}\right) \end{equation}$$

$$\begin{equation} e^{i x} = \cos x + j \sin x \end{equation}$$