Proof of One minus Cosine double angle identity

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

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

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

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

It is your turn to learn how to derive the one minus cosine of double angle is mathematically equal to the two times the sine squared of angle in trigonometry.

If the symbol theta is considered to represent an angle of a right triangle, then the square of sine of angle is written as $\sin^2{\theta}$ and the cosine of double angle is written as $\cos{2\theta}$ in trigonometric mathematics.

Express Difference of One and cos double angle

The subtraction of cosine of double angle from one is written in mathematical form by writing one and cos double angle in a row with a minus sign between them.

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

Expand the cosine of double angle function

As per the cos double angle identity, the cosine of double angle function can be expanded in terms of sine squared of angle.

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

Simplify the Trigonometric expression

For evaluating the one minus cosine of double angle, the trigonometric expression at the right hand side of the equation should be simplified.

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

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

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

It has derived that the one minus cosine of double angle is equal to the two times the square of sine of angle. It is called the one minus cosine double angle formula.

Thus, we can prove this trigonometric identity in terms of either $A$ or $x$ or any other symbol by the above procedure.