$\large 1+\cos 2\theta = 2\cos^2 \theta$
As per the expansion of the cosine of double angle in terms of cosine of angle.
$\cos 2 \theta = 2\cos^2 \theta -1$
$\therefore \,\,\,\,\,\,$ $1+\cos 2\theta = 2\cos^2 \theta$
It is proved that the addition of number one and cosine of double angle is equal to twice the cosine squared angle.
The fundamental trigonometric identity can be written as follows if the angle of the right angled triangle is $x$.
$1+\cos 2x = 2\cos^2 x$
The angle of the right angled triangle can be denoted by any symbol and the addition of $1$ and cosine of double angle is expressed as twice the square of cosine of angle.