Vtyjkyuk

I am trying to understand why $$cos2 theta=cos^2theta-sin^2theta$$

I cannot find any information on this identity on the Wikipedia page listing the trigonometric identities

It comes from the more general formula $$ cos ( x+y) = cos (x) cos (y) - sin (x) sin(y)$$

If you let x=y, you will get $$ cos ( x+x) = cos (x) cos (x) - sin (x) sin(x)$$

Which is the same as $$cos2 x=cos^2 x-sin^2 x$$

The linked wikipedia page gives the standard angle-addition formula for cosine,

$cos( alpha pm beta) = cos alpha cos beta mp sin alpha sin beta, tag 1$

and provides a demonstration that it is so; given (1), we have

$cos 2 theta = cos (theta + theta) = cos theta cos theta - sin theta sin theta = cos^2 theta - sin^2 theta; tag 2$

so that's one way to arrive at the desired formula.

I prefer to us the matrix exponential:

$e^i psi = cos psi + i sin psi, ; psi in Bbb R; tag 2$

then

$cos(psi + phi) + i sin (psi + phi) = e^i(psi + phi) = e^ipsie^iphi$
$= (cos psi + i sin psi)(cos phi + i sin phi)$
$= cos psi cos phi - sin psi sin phi + i(sin psi cos phi + sin phi cos psi); tag 3$

equating the real parts on either side of (3) yields

$cos(psi + phi) = cos psi cos phi - sin psi sin phi, tag 4$

essentially the same as (1), easily seen to lead to (2) by taking $theta = psi = phi$.