# Sine squared Power reduction identities

## Formulae

$(1).\,\,\,$ $\sin^2{\theta} \,=\, \dfrac{1-\cos{(2\theta)}}{2}$

$(2).\,\,\,$ $\sin^2{\Bigg(\dfrac{\theta}{2}\Bigg)} \,=\, \dfrac{1-\cos{\theta}}{2}$

A trigonometric identity that expresses the reduction of square of sine function in terms of cosine is called the power reduction identity of sine squared function.

### Introduction

There are two popular sine squared power reducing identities and they are used as formulas in trigonometry.

Let theta be an angle of a right triangle. The double angle and half angles are written as $2\theta$ and $\dfrac{\theta}{2}$ respectively. The square of sine of angle and cosine of angle are written in mathematical form as $\sin^2{\theta}$ and $\cos{\theta}$ respectively. The cosine of double angle is written as $\cos{2\theta}$. Similarly, the sine squared of half angle in mathematical form is written as $\sin^2{\Big(\dfrac{\theta}{2}\Big)}$.

Now, the power reducing trigonometric identities in terms of the sine squared functions are written mathematically in the following two forms in trigonometry.

#### Angle to Double angle form

$\sin^2{\theta} \,=\, \dfrac{1-\cos{(2\theta)}}{2}$

The sine squared of angle is equal to the quotient of one minus cos of double angle by two.

#### Half angle to Angle form

$\sin^2{\Bigg(\dfrac{\theta}{2}\Bigg)} \,=\, \dfrac{1-\cos{\theta}}{2}$

The sine squared of half angle is equal to the quotient of one minus cosine of angle by two.

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