The cosine double angle identity can be expanded in terms of square of cosine function as follows.

$\cos{2\theta}$ $\,=\,$ $2\cos^2{\theta}-1$

Now, you can know how to prove the cosine double angle identity in terms of cos squared function in trigonometry.

Let theta denotes an angle of a right triangle (or right angled triangle), the cosine in square form is written as $\cos^2{\theta}$ and cosine of double angle function is written as $\cos{2\theta}$.

As per the proof of cos double identity, the cosine of double angle is equal to the difference of the squares of cosine and sine of angle.

$\cos{2\theta}$ $\,=\,$ $\cos^2{\theta}-\sin^2{\theta}$

According to the sine squared identity, the square of sine function can be written in terms of square of cosine function.

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

We can simplify the right hand side trigonometric expression of the equation for deriving the expansion of cosine double angle rule in square of cosine function.

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

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

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

The angle in the cos double angle identity can be denoted by any symbol. However, it is written popularly in the below two forms. You can prove this trigonometric identity in terms of any symbol by the above steps.

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

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

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