sin^2(x) + cos^2(x) = 1

sin(u + v) = sin u cos v + cos u sin v

cos(u + v) = cos u cos v - sin u sin v

Flip-and-shift-pi/2-units identities

(or the cofunction identities)

sin(pi / 2 - x) = cos(x)        cos(pi / 2 - x) = sin(x)

tan(pi / 2 - x) = cot x          cot(pi / 2 - x) = tan x

sec(pi / 2 - x) = csc x        csc(pi / 2 - x) = sec x

Shifting left pi/2 gives you

sin(x+pi/2) = cos(x)        cos(x+pi/2) = -sin(x)

From sin(u+v) = sin u cos v + cos u sin v let u = x and v = x to get

From cos(u+v) = cos u cos v - sin u sin v let u = x and v = x to get

(Last two steps: substitute 1 - sin^2(x) for cos^2(x) then in cos^2(x) - sin^2(x) , then substitute 1 - cos^2(x) for sin^2(x) in cos^2(x) - sin^2(x) )

From tan(u+v) = (tan u + tan v) / ( 1 - tan u tan v) substitute u = x and v = x to get

From cos(2 u) = 1 - 2 sin^2(u), let u = x/2 to get sin(x) = 1 - 2 sin^2(x/2), which you solve for sin(x/2) to get

From cos(2 u) = 2 cos^2(u) - 1, let u = x/2 to get cos(x) = 2 cos^2(x/2) - 1, which you solve for cos(x/2) to get

Since tan(x/2) = sin(x/2) / cos(x/2), use the preceding two identities to get