trigonometric formulas

Sum to product

\sin α + \sin β = 2 \sin (\frac{α + β}{2}) \cos (\frac{α - β}{2})

\sin α - \sin β = 2 \sin (\frac{α - β}{2}) \cos (\frac{α + β}{2})

\cos α - \cos β = - 2 \sin (\frac{α + β}{2}) \sin (\frac{α - β}{2})

\cos α + \cos β = 2 \cos (\frac{α + β}{2}) \cos (\frac{α - β}{2})

\tanh (x) = - i \tan (i x)

\arctan (x) = \frac{1}{i} arctanh (i x)

\arctan (i x) = i arctanh (x)

i arccot (- i x) = - arctanh (\frac{1}{x})

i arccoth (i x) = arctan (\frac{1}{x})

\arctan (z) = \frac{i}{2} \ln (\frac{1 - i z}{1 + i z})

\sinh (i z) = i \sin (z)

\arcsin (x) = - i arsinh (i x)

\arctan (a) + \arctan (b) = {\begin{cases} \arctan (\frac{a + b}{1 - a b}) & if a b < 1, \\ \arctan (\frac{a + b}{1 - a b}) + π & if a b > 1, \\ \frac{π}{2} & if a b = 1 and a, b > 0, \\ - \frac{π}{2} & if a b = 1 and a, b < 0. \end{cases}

arctanh (a) + arctanh (b) = arctanh (\frac{a + b}{1 + a b}) if | a b | < 1

artanh (x) = \frac{1}{2} \ln (\frac{1 + x}{1 - x})