Math Doubts

Cosine squared Power reduction identities

Formulae

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

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

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

Introduction

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

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

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

Angle to Double angle form

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

The cosine squared of angle is equal to the quotient of one plus cos of double angle by two.

Half angle to Angle form

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

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

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