Trigonometry identities Power Reduction Calculator
The Trigonometry Identities Power Reduction Calculator computes sin2u, cos2u and tan2u for given angle
using following formulas:
sin2u = 1/2 - (1/2)cos(2u))cos2u = 1/2 + (1/2)cos(2u))
tan2u = (1 - cos(2u)) / (1 + cos(2u))
sin3u = (3/4)sinu - (1/4)sin(3u)
cos3u= (3/4)cosu + (1/4)cos(3u) and so on ..
Proof the power reduction formula for sin and cosin
Proof for sin :
cos(2u) = cos2u - sin2u ............(1)
we will use the Pythagorean Identities as sin2u + cos2u = 1so cos2u = 1 - sin2u, we can substitute the value of cos2u in equation (1) and we will get as :
cos2u = (1 - sin2u) - sin2u
cos2u = (1 - 2sin2u)
Now sustract 1 from the both sides
cos2u - 1 = (1 - 2sin2u) - 1
2sin2u = 1 - cos2u
sin2u = (1 - cos2u)/2
sin2u = 1/2(1 - cos2u)
Proof for cos :
cos(2u) = cos2u - sin2u ............(1)
we will use the Pythagorean Identities as sin2u + cos2u = 1so sin2u = 1 -cos2u , we can substitute the value of sin2u in equation (1) and we will get as :
cos2u = cos2u - (1 -cos2u)
cos2u = 2cos2u - 1
Now add 1 from the both sides
cos2u + 1 = 2cos2u - 1 + 1
cos2u + 1 = 2cos2u
2cos2u = cos2u + 1
cos2u = 1/2(cos2u + 1)