Interstingly, all the main Trigonometric functions can be defined in terms of sine and the square root function.
cos θ = 1 − sin 2 θ {\displaystyle \cos \theta ={\sqrt {1-\sin ^{2}\theta }}}
tan θ = sin θ 1 − sin 2 θ {\displaystyle \tan \theta ={\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}}
cot θ = 1 − sin 2 θ sin θ {\displaystyle \cot \theta ={\frac {\sqrt {1-\sin ^{2}\theta }}{\sin \theta }}}
sec θ = 1 1 − sin 2 θ {\displaystyle \sec \theta ={\frac {1}{\sqrt {1-\sin ^{2}\theta }}}}
csc θ = 1 sin θ {\displaystyle \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 ( x 1 − x 2 ) {\displaystyle \arcsin(x)=\arctan \left({\frac {x}{\sqrt {1-x^{2}}}}\right)}
arccos ( x ) = arctan ( 1 − x 2 x ) {\displaystyle \arccos(x)=\arctan \left({\frac {\sqrt {1-x^{2}}}{x}}\right)}
arcsec ( x ) = arctan ( x 2 − 1 ) {\displaystyle \operatorname {arcsec}(x)=\arctan \left({\sqrt {x^{2}-1}}\right)}
arccsc ( x ) = arctan ( 1 x 2 − 1 ) {\displaystyle \operatorname {arccsc}(x)=\arctan \left({\frac {1}{\sqrt {x^{2}-1}}}\right)}
arccot ( x ) = arctan ( 1 x ) {\displaystyle \operatorname {arccot}(x)=\arctan \left({\frac {1}{x}}\right)}
