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