How do I prove that $\cos\left(2x\right)=1-2\sin^2\left(x\right)$? While trying to solve the equation $\sin\left(x\right)=\cos\left(2x\right)$, a user on this forum suggested that I turn the equation into a quadratic form by converting $\cos(2x)$ using the identity $\cos\left(2x\right)=1-2\sin^2\left(x\right)$.
What is the logic behind this identity and how can I derive it?

write $$\cos (2x)=\cos(x+x)$$
you know $$\cos(x+y)=\cos x\cdot \cos y-\sin x\cdot \sin y$$
So, $$\cos(2x)=\cos x\cdot \cos x-\sin x\cdot \sin x$$
Or, $$\cos(2x)=\cos^2x-\sin^2x$$
write $\cos^2x$ as $1-\sin^2x$ 
So, $$\cos(2x)=1-\sin^2x-\sin^2x$$

you get$$\cos(2x)=1-2\sin^2x$$ 

Similarly get $$\cos(2x)=\cos^2x-\sin^2x=1-2\sin^2x=2\cos^2x-1$$
