Proof of One plus Cosine double angle identity

The one plus cosine double angle identity can be written in terms of any symbol but it is popularly written in the following three forms.

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

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

$(3).\,\,\,$ $1+\cos{(2A)}$ $\,=\,$ $2\cos^2{A}$

Now, let’s learn how to prove the one plus cosine of double angle is equal to the two times the cosine squared of angle mathematically in trigonometry.

When the symbol theta denotes an angle of a right triangle, the square of cosine of angle is written as $\cos^2{\theta}$ and the cosine of double angle is written as $\cos{2\theta}$ in trigonometry.

Express Sum of One and cosine double angle

The addition of one plus cosine of double angle is written in mathematics by expressing one and cos double angle in a row with a plus sign between them.

$1+\cos{(2\theta)}$

Expand the cosine of double angle function

According to the cos double angle identity, the cosine of double angle function can be expanded in terms of cosine of angle.

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

Simplify the Trigonometric expression

Now, simplify the trigonometric expression at the right hand side of the equation for evaluating the one plus cos double angle function in mathematical form.

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

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

$\implies \require{cancel}$ $1+\cos{(2\theta)}$ $\,=\,$ $\cancel{1}-\cancel{1}+2\cos^2{\theta}$

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

Mathematically, it is proved that the one plus cosine of double angle is equal to the two times the square of cosine of angle. It is called the one plus cosine double angle rule.

In this way, you can prove this trigonometric identity in terms of either $x$ or $A$ or any other symbol by following the same procedure.