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.

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

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

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

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

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