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Sine squared Power reduction identities


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

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

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.


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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