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