User:Hal Canary/Draft of Trig Identities

Interstingly, all the main Trigonometric functions can be defined in terms of sine and the square root function.

$$\cos \theta = \sqrt{1-\sin ^2 \theta }$$

$$\tan \theta = \frac{\sin \theta}{\sqrt{1-\sin ^2 \theta}}$$

$$\cot \theta = \frac{\sqrt{1-\sin ^2 \theta}}{\sin \theta}$$

$$\sec \theta = \frac{1}{\sqrt{1-\sin ^2 \theta }}$$

$$\csc \theta = \frac{1}{\sin \theta }$$

Similarly, the inverse trigonometric functions can easily be defined in terms of one of them---for example, the arctangent function.

$$\arcsin(x) = \arctan\left({\frac{x}{\sqrt{1 - x^2}}}\right)$$

$$\arccos(x) = \arctan\left({\frac{\sqrt{1-x^2}}{x}}\right)$$

$$\arcsec(x) = \arctan\left({\sqrt{x^2-1}}\right)$$

$$\arccsc(x) = \arctan\left({\frac{1}{\sqrt{x^2-1}}}\right)$$

$$\arccot(x) = \arctan\left({\frac{1}{x}}\right)$$