# Sub multiple angle identities

In some cases, it is essential to express trigonometric functions in terms of sum multiple angle trigonometric functions to find their values mathematically. So, learn how to expand trigonometric functions in terms of trigonometric functions which contain sub multiple angles. The following submultiple angle identities are used as formulae in trigonometric mathematics but they are similar to multiple angle formulas.

### Half angle formulas

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

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

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

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

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

### One third angle formulas

Learn how to expand trigonometric functions in terms of one third angle trigonometric functions.

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

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

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

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

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