Double-angle formulas can be expanded to multiple-angle functions (triple, quadruple, quintuple, and so on) by using the angle sum formulas, and then reapplying the double-angle formulas.
\(\begin{array}{l}\large sin(A+B)=sinA\;cosB+cosA\;sinB\end{array} \)
\(\begin{array}{l}\large sin(A-B)=sinA\;cosB-cosA\;sinB\end{array} \)
\(\begin{array}{l}\large cos(A+B)=cosA\;cosB-sinA\;sinB\end{array} \)
\(\begin{array}{l}\large cos(A-B)=cosA\;cosB+sinA\;sinB\end{array} \)
\(\begin{array}{l}\large sin\alpha +sin\beta =2sin\frac{\alpha +\beta }{2}cos\frac{\alpha -\beta }{2}\end{array} \)
\(\begin{array}{l}\large sin\alpha -sin\beta =2sin\frac{\alpha -\beta }{2}cos\frac{\alpha +\beta }{2}\end{array} \)
\(\begin{array}{l}\large cos\alpha +cos\beta =2cos\frac{\alpha +\beta }{2}cos\frac{\alpha -\beta }{2}\end{array} \)
\(\begin{array}{l}\large cos\alpha -cos\beta =-2sin\frac{\alpha +\beta }{2}sin\frac{\alpha -\beta }{2}\end{array} \)
\(\begin{array}{l}\large sin2\alpha =2\;sin\alpha\;cos\alpha\end{array} \)
\(\begin{array}{l}\large cos2\alpha =cos^{2}\alpha -sin^{2}\alpha = 2cos^{2}\alpha -1=1-2sin^{2}\alpha\end{array} \)
\(\begin{array}{l}\large tan2\alpha =\frac{2tan\alpha }{1-tan^{2}\alpha }\end{array} \)
Half Angle Formulas
\(\begin{array}{l}\large sin\left ( \frac{a}{2} \right )=\pm \sqrt{\frac{(1-cos\;a)}{2}}\end{array} \)
\(\begin{array}{l}\large cos\left ( \frac{a}{2} \right )=\pm \sqrt{\frac{(1+cos\;a)}{2}}\end{array} \)
\(\begin{array}{l}\large tan\left ( \frac{a}{2} \right )=\frac{1-cos\;a}{sin\;a}=\frac{sin\;a}{1+cos\;a}\end{array} \)
These formulas can also be written as:
\(\begin{array}{l}\large sin\left ( \frac{a}{2} \right )=\sqrt{\frac{1-cos(a)}{2}}\end{array} \)
\(\begin{array}{l}\large cos\left ( \frac{a}{2} \right )=\sqrt{\frac{1+cos(a)}{2}}\end{array} \)
\(\begin{array}{l}\large tan\left ( \frac{a}{2} \right )=\sqrt{\frac{1-cos(a)}{1+cos(a)}}\end{array} \)
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