The trigonometric functions are often appeared with multiple angles. It is not possible to find their values directly but their values can be evaluated by expressing each trigonometric function in its expansion form. Now, learn how to expand trigonometric functions with multiple angles. The following multiple angle identities are used as formulae in mathematics.

Learn how to expand double angle trigonometric functions in terms of trigonometric functions.

$(1)\,\,\,\,$ $\sin{2\theta}$ $\,=\,$ $2\sin{\theta}\cos{\theta}$

$(2)\,\,\,\,$ $\cos{2\theta}$ $\,=\,$ $\cos^2{\theta}-\sin^2{\theta}$

$(3)\,\,\,\,$ $\tan{2\theta}$ $\,=\,$ $\dfrac{2\tan{\theta}}{1-\tan^2{\theta}}$

$(4)\,\,\,\,$ $\cot{2\theta}$ $\,=\,$ $\dfrac{\cot^2{\theta}-1}{2\cot{\theta}}$

Learn how to expand triple angle trigonometric functions in terms of trigonometric functions.

$(1)\,\,\,\,$ $\sin{3\theta}$ $\,=\,$ $3\sin{\theta}-4\sin^3{\theta}$

$(2)\,\,\,\,$ $\cos{3\theta}$ $\,=\,$ $4\cos^3{\theta}-3\cos{\theta}$

$(3)\,\,\,\,$ $\tan{3\theta}$ $\,=\,$ $\dfrac{3\tan{\theta}-\tan^3{\theta}}{1-3\tan^2{\theta}}$

$(4)\,\,\,\,$ $\cot{3\theta}$ $\,=\,$ $\dfrac{3\cot{\theta}-\cot^3{\theta}}{1-3\cot^2{\theta}}$

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