Proof of Cosine squared Power reduction identity
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According to the cosine squared power reduction identity, the square of cosine of angle is equal to one plus cos of double angle by two. It can be expressed in any one of the following forms popularly.
$(1).\,\,\,$ $\cos^2{\theta} \,=\, \dfrac{1+\cos{(2\theta)}}{2}$
$(2).\,\,\,$ $\cos^2{x} \,=\, \dfrac{1+\cos{(2x)}}{2}$
$(3).\,\,\,$ $\cos^2{A} \,=\, \dfrac{1+\cos{(2A)}}{2}$
Let a symbol theta be an angle of a right triangle (right angled triangle), the squares of cosine and sine functions are written as $\cos^2{\theta}$ and $\sin^2{\theta}$ respectively. In the same way, the cosine of double angle is written as $\cos{2\theta}$ mathematically in trigonometry.
Relation between functions in square form
As per the Pythagorean identity, the sum of squares of sine and cosine functions is equal to one.
$\implies$ $\cos^2{\theta}+\sin^2{\theta} = 1$
Express the equation in Cos of double angle
The above trigonometric equation can be transformed in terms of cosine of double angle by an acceptable setting.
$\implies$ $\cos^2{\theta}$ $\,=\,$ $1-\sin^2{\theta}$
Now, add the square of cosine of angle to both sides of the trigonometric equation.
$\implies$ $\cos^2{\theta}+\cos^2{\theta}$ $\,=\,$ $1-\sin^2{\theta}+\cos^2{\theta}$
$\implies$ $2\cos^2{\theta}$ $\,=\,$ $1+\cos^2{\theta}-\sin^2{\theta}$
The subtraction of second and third terms in the right hand side of the equation represents the expansion of cosine of double angle trigonometric identity.
$\implies$ $2\cos^2{\theta}$ $\,=\,$ $1+\cos{2\theta}$
Express Cos squared function in Cos of Double angle
The equation can be simplified to write the cosine squared of angle in terms of cosine of double angle function.
$\implies$ $2\cos^2{\theta} = 1+\cos{(2\theta)}$
$\,\,\,\therefore\,\,\,\,\,\,$ $\cos^2{\theta} = \dfrac{1+\cos{(2\theta)}}{2}$
It expresses that the square of cosine of angle is reduced by expressing it as quotient of one plus cosine of double angle by two and it is called the power reduction trigonometric identity of cosine squared function.
