# 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 ..

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)