### Trigonometry identities Power Reduction Calculation

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 ..

The power reduction formulas allows to transform sin

^{2}(u) and cos

^{2}(u) into expressions that contains the first power of cosine of double argument. These functions are in the same way as double-angle and half-angle functions. The power reduction formulas can be derived through the use of double-angle, half-angle formulas and Pythagorean Identity. The use of a power reduction formula expresses the quantity without the exponent.

**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 = 1

so 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 = 1

so 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)