Exsecant

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \theta }}\operatorname {exsec} (\theta )=\tan(\theta )\sec(\theta )={\frac {\sin(\theta )}{(\cos(\theta ))^{2}}}}$[21]

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \theta }}\operatorname {excsc} (\theta )=-\cot(\theta )\csc(\theta )={\frac {-\cos(\theta )}{(\sin(\theta ))^{2}}}}$

${\displaystyle \int \operatorname {exsec} (\theta )\,\mathrm {d} \theta =\ln \left[\cos \left({\frac {\theta }{2}}\right)+\sin \left({\frac {\theta }{2}}\right)\right]-\ln \left[\cos \left({\frac {\theta }{2}}\right)-\sin \left({\frac {\theta }{2}}\right)\right]-\theta +C}$[21]

${\displaystyle \int \operatorname {excsc} (\theta )\,\mathrm {d} \theta =\ln \left[\tan \left({\frac {\theta }{2}}\right)\right]-\theta +C}$

The inverse functions arcexsecant[26] (arcexsec,[5][26] aexsec,[32][33] aexs, exsec⁻¹) and arcexcosecant (arcexcosec, arcexcsc,[5] aexcsc, aexc, arccoexsecant, arccoexsec, excsc⁻¹) exist as well:

${\displaystyle \operatorname {arcexsec} (y)=\operatorname {arcsec}(y+1)=\arccos \left({\frac {1}{y+1}}\right)=\arctan({\sqrt {y^{2}+2y}})}$[26][32][33] (for y ≤ −2 or y ≥ 0)[26]

${\displaystyle \operatorname {arcexcsc} (y)=\operatorname {arccsc}(y+1)=\arcsin \left({\frac {1}{y+1}}\right)\,}$

Derived from the unit circle:

${\displaystyle \operatorname {exsec} (\theta )=\sec(\theta )-\cos(\theta )-\operatorname {versin} (\theta ).}$

${\displaystyle \operatorname {excsc} (\theta )=\operatorname {csc} (\theta )-\sin(\theta )-\operatorname {coversin} (\theta ).}$

The exsecant function is related to the tangent function by

${\displaystyle \operatorname {exsec} (\theta )=\tan(\theta )\tan \left({\frac {\theta }{2}}\right).}$[23]

In analogy, the excosecant function is related to the cotangent function by

${\displaystyle \operatorname {excsc} (\theta )=\cot(\theta )\cot \left({\frac {\theta }{2}}\right).}$

The exsecant function is related to the sine function by

${\displaystyle \operatorname {exsec} (\theta )={\frac {1}{\sqrt {1-(\sin(\theta ))^{2}}}}-1.}$

In analogy, the excosecant function is related to the cosine function by

${\displaystyle \operatorname {excsc} (\theta )={\frac {1}{\sqrt {1-(\cos(\theta ))^{2}}}}-1.}$[30]

The exsecant and excosecant functions can be extended into the complex plane.[21]

${\displaystyle \lim _{\theta \to 0}{\frac {\operatorname {exsec} (\theta )}{\theta }}=0}$[5]

${\displaystyle {\frac {\operatorname {versin} (\theta )+\operatorname {coversin} (\theta )}{\operatorname {versin} (\theta )-\operatorname {coversin} (\theta )}}-{\frac {\operatorname {exsec} (\theta )+\operatorname {excsc} (\theta )}{\operatorname {exsec} (\theta )-\operatorname {excsc} (\theta )}}={\frac {2\operatorname {versin} (\theta )\operatorname {coversin} (\theta )}{\operatorname {versin} (\theta )-\operatorname {coversin} (\theta )}}}$[5]

${\displaystyle (\operatorname {exsec} (\theta )+\operatorname {versin} (\theta ))\,(\operatorname {excsc} (\theta )+\operatorname {coversin} (\theta ))=\sin(\theta )\cos(\theta )}$[5]

${\displaystyle \operatorname {exsec} (2\theta )={\frac {2\sin ^{2}(\theta )}{1-2\sin ^{2}(\theta )}}}$[5]

${\displaystyle \operatorname {exsec} (2\theta )\,\cos(2\theta )=\tan(\theta )}$[5]

${\displaystyle \operatorname {exsec} ^{2}(\theta )+2\operatorname {exsec} (\theta )=\tan ^{2}(\theta )}$[5]