Trigonometric Identities and Differential Equations

Trigonometric Identities
\begin{aligned}  \sin(A\pm B)&=\sin A\cos B\pm\cos A\sin B\\  \cos(A\pm B)&=\cos A\cos B\mp\sin A\sin B\\  \tan(A\pm B)&=\frac{\tan A\pm \tan B}{1\mp \tan A\tan B}  \end{aligned}
\begin{aligned}  \sin A+\sin B&=2\sin\frac{1}{2}(A+B)\cos\frac{1}{2}(A-B)\\  \sin A-\sin B&=2\cos\frac{1}{2}(A+B)\sin\frac{1}{2}(A-B)\\  \cos A+\cos B&=2\cos\frac{1}{2}(A+B)\cos\frac{1}{2}(A-B)\\  \cos A-\cos B&=-2\sin\frac{1}{2}(A+B)\sin\frac{1}{2}(A-B)  \end{aligned}

Reduction to Separable Form
Equations of the form
y'=g(\frac{y}{x}).
Set \frac{y}{x}=v. Then y=vx and y'=v+xv'. Thus the original equation becomes v+xv'=g(v) which is separable.

Linear Change of Variable
y'=f(ax+by+c) can be solved by setting u=ax+by+c.

Newton’s Law of Cooling
Rate of change of temperature of an object (\frac{dT}{dt}) is proportional to the difference between its own temperature (T) and the ambient temperature (T_\text{env}). That is, \frac{dT}{dt}=-k(T-T_\text{env}).

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