Proof of Sine squared Power reduction identity

According to the sine squared power reduction identity, the square of sine of angle is equal to one minus cos of double angle by two. It is mathematically expressed in any one of the following forms popularly.

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

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

$(3).\,\,\,$ $\sin^2{A} \,=\, \dfrac{1-\cos{(2A)}}{2}$

When the angle of a right triangle is denoted by a symbol theta, the squares of sine and cosine functions are written as $\sin^2{\theta}$ and $\cos^2{\theta}$ respectively. Similarly, the cosine of double angle is written as $\cos{2\theta}$ in mathematical form in trigonometry.

Relation between functions in square form

According to the Pythagorean identity, the sum of squares of sine and cosine functions is equal to one.

$\cos^2{\theta}+\sin^2{\theta} = 1$

Express the equation in Cos of double angle

The trigonometric equation can be converted into cosine of double angle by an acceptable setting.

$\implies$ $\cos^2{\theta} = 1-\sin^2{\theta}$

Now, add sine squared of angle to both sides of the trigonometric equation and it helps us to express the one side of equation as cosine of double angle function.

$\implies$ $\cos^2{\theta}-\sin^2{\theta} = 1-\sin^2{\theta}-\sin^2{\theta}$

$\implies$ $\cos^2{\theta}-\sin^2{\theta} = 1-2\sin^2{\theta}$

According to the cosine of double angle trigonometric identity, the subtraction of square of sine from square of cosine is equal to the cosine of double angle.

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

Express Sine squared in Cosine of Double angle

The trigonometric equation can be simplified for expressing the sine squared of angle in terms of cosine of double angle function.

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

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

This equation expresses the square of sine of angle is reduced by expressing it as cosine of double angle and it is called the power reduction trigonometric identity of sine squared function.