To solve this equation, we will first rewrite ctgx as 1/tgx and expand the left side using trigonometric identities:

cosx * (1/tgx) - sinx = cos2x
cosx/tgx - sinx = cos2x

Next, we will replace tgx with sinx/cosx to simplify the equation:

cosx/(sinx/cosx) - sinx = cos2x
cos^2(x)/sinx - sinx = cos2x

Now, we will multiply both sides by sinx to eliminate the denominator:

cos^2(x) - sin^2(x) = sinx*cos2x

Using the Pythagorean identity cos^2(x) - sin^2(x) = cos(2x), the equation becomes:

cos(2x) = sinx*cos2x

Therefore, cosx ctgx - sinx = cos2x is equivalent to cos(2x) = sinxcos2x.