Forward-inverse trigonometric formulae

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A "forward-inverse" trigonometric function is a function of the form foo(arcbar(x)) where foo(x) and bar(x) are trigonometric functions (one of sine, cosine, tangent, cosecant, secant, cotangent).

arcbar(x) (or equivalently bar-1(x)) is the inverse function of bar(x). So foo(arcbar(x)) is foo of the inverse of bar, hence "forward-inverse".

The tables below list these identities for every combination of trigonometric formulae for both circular and hyperbolic trigonometric functions. Derivations of these formulae are detailed in hidden sections in order to remove clutter, but you can expand these hidden section to read them by clicking on the "show" tags.


Circular functions

Forward-inverse circular trigonometric formulae
foo(arcbar(x)) arcsin arccos arctan arccosec arcsec arccot
sin x \sqrt{1-x^2} x/\sqrt{1+x^2} 1/x \sqrt{x^2-1}/x 1/\sqrt{1+x^2}
cos \sqrt{1-x^2} x 1/\sqrt{1+x^2} \sqrt{x^2-1}/x 1/x x/\sqrt{1+x^2}
tan x/\sqrt{1-x^2} \sqrt{1-x^2}/x x 1/\sqrt{x^2-1} \sqrt{x^2-1} 1/x
cosec 1/x 1/\sqrt{1-x^2} \sqrt{1+x^2}/x x x/\sqrt{x^2-1} \sqrt{1+x^2}
sec 1/\sqrt{1-x^2} 1/x \sqrt{1+x^2} x/\sqrt{x^2-1} x \sqrt{1+x^2}/x
cot \sqrt{1-x^2}/x x/\sqrt{1-x^2} 1/x \sqrt{x^2-1} 1/\sqrt{x^2-1} x


Hyperbolic functions

Forward-inverse hyperbolic trigonometric formulae
foo(arcbar(x)) arcsinh arccosh arctanh arccosech arcsech arccoth
sinh x \sqrt{x^2-1} x/\sqrt{1-x^2} 1/x \sqrt{1-x^2}/x 1/\sqrt{x^2-1}
cosh \sqrt{1+x^2} x 1/\sqrt{1-x^2} \sqrt{1+x^2}/x 1/x x/\sqrt{x^2-1}
tanh x/\sqrt{1+x^2} \sqrt{x^2-1}/x x 1/\sqrt{1+x^2} \sqrt{1-x^2} 1/x
cosech 1/x 1/\sqrt{x^2-1} \sqrt{1-x^2}/x x x/\sqrt{1-x^2} \sqrt{x^2-1}
sech 1/\sqrt{1+x^2} 1/x \sqrt{1-x^2} x/\sqrt{1+x^2} x \sqrt{x^2-1}/x
coth \sqrt{1+x^2}/x x/\sqrt{x^2-1} 1/x \sqrt{1+x^2} 1/\sqrt{1-x^2} x