# Proof of Addition of 1 and Cos double angle formula

Take, $x$ is a variable but represents an angle of a right triangle, $2x$ is a double angle but an angle of another right triangle. Therefore, the squares of sine and cosine functions are written as $\sin^2{x}$ and $\cos^2{x}$ respectively, and the cosine of double angle function is expressed as $\cos{2x}$ in mathematical form.

Now, let us start deriving the trigonometric identity that helps us to evaluate the sum of one and cosine of double angle function.

### Addition of One and cos double angle

Add the cos double angle function to number one for expressing the addition in mathematical form.

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

### Expand the cosine of double angle function

As per the cos double angle identity, the $\cos{2x}$ function can be expanded in terms of sine and cosine of angle.

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

### Simplify the trigonometric expression

Now, let’s simplify the trigonometric expression in the right-hand side of the trigonometric equation.

$=\,\,\,$ $1+\cos^2{x}-\sin^2{x}$

$=\,\,\,$ $1-\sin^2{x}+\cos^2{x}$

As per the Pythagorean identity of sine and cosine functions, the subtraction of sine squared of angle from one is equal to cosine squared of angle.

$=\,\,\,$ $\cos^2{x}+\cos^2{x}$

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

Therefore, it is proved that the sum of one and cosine of double angle is equal to two times the cos squared of angle. If you want to express this proof for a trigonometric expression like either $1+\cos{2\theta}$ or $1+\cos{2A}$, then take $\theta$ or $A$ instead of $x$.

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