Hyperbolic functions
Sums of arguments

sinh (x + y) = sinh x cosh y + cosh x sinh y cosh (x + y) = cosh x cosh y + sinh x sinh y tanh (x + y) = tanh x + tanh y 1 + tanh x tanh y {\displaystyle {\begin{aligned}\sinh(x+y)&=\sinh x\cosh y+\cosh x\sinh y\\\cosh(x+y)&=\cosh x\cosh y+\sinh x\sinh y\\\tanh(x+y)&={\frac {\tanh x+\tanh y}{1+\tanh x\tanh y}}\\\end{aligned}}} particularly cosh (2 x) = sinh 2 x + cosh 2 x = 2 sinh 2 x + 1 = 2 cosh 2 x − 1 sinh (2 x) = 2 sinh x cosh x tanh (2 x) = 2 tanh x 1 + tanh 2 x {\displaystyle {\begin{aligned}\cosh(2x)&=\sinh ^{2}{x}+\cosh ^{2}{x}=2\sinh ^{2}x+1=2\cosh ^{2}x-1\\\sinh(2x)&=2\sinh x\cosh x\\\tanh(2x)&={\frac {2\tanh x}{1+\tanh ^{2}x}}\\\end{aligned}}}

Also: sinh x + sinh y = 2 sinh (x + y 2) cosh (x − y 2) cosh x + cosh y = 2 cosh (x + y 2) cosh (x − y 2) {\displaystyle {\begin{aligned}\sinh x+\sinh y&=2\sinh \left({\frac {x+y}{2}}\right)\cosh \left({\frac {x-y}{2}}\right)\\\cosh x+\cosh y&=2\cosh \left({\frac {x+y}{2}}\right)\cosh \left({\frac {x-y}{2}}\right)\\\end{aligned}}}
oulipo:avoid_letter[t]