Trigonometric Identities 11Oct Basic Definitions tan x=sin x+cos x csc u = \frac{1}{sin u} sec u = \frac{1}{cos u} cot u = \frac{1}{tan u} Pythagorean Identities sin^2u+cos^2u=1 1+tan^2u=sec^2u 1+cot^2u=csc^2u Co-Function Identities sin(u\pm\frac{\pi}{2}) = cos u csc(u\pm\frac{\pi}{2}) = sec u tan(u\pm\frac{\pi}{2}) = cot u Sum-Difference Formulas sin(u\pm v)=sinucosv\pm cos u sin v cos(u\pm v)=cosucosv\mp sin u sin v tan(u\pm v)=\frac{tan u\pm tan v}{1\mp tan u tan v} Double Angle Formulas sin(2u)=2sin u cos u cos(2u)=cos^2u-sin^2u tan(2u)=\frac{2tan u}{1-tan^2u} Power-Reducing/Half Angle Formulas sin^2u=\frac{1-cos(2u)}{2} cos^2u=\frac{1+cos(2u)}{2} tan^2u=\frac{1-cos(2u)}{1+cos(2u)} Sum-to-Product Formulas sin(u) \pm sin(v) = 2sin(\frac{u\pm v}{2})cos(\frac{u\mp v}{2}) cos(u) + cos(v) = 2cos(\frac{u+v}{2})cos(\frac{u-v}{2}) cos(u) - cos(v) = -2sin(\frac{u+v}{2})sin(\frac{u-v}{2}) Product-to-Sum Formulas sinusinv = \frac{1}{2}[cos(u-v)-cos(u+v)] cosucosv = \frac{1}{2}[cos(u-v)+cos(u+v)] sinucosv = \frac{1}{2}[sin(u+v)+sin(u-v)] Hyperbolic cosh^2u-sinh^2u=1 1-tanh^2u=sech^2u sinh2u=cosh^2u+sinh^2u