The question was
Prove $$\frac{1+\sin2x+\cos2x}{\cos x+\sin x}=2\cos x$$
I simplified it using several trigonometric identities, what I got is this "$\dfrac{2\cos^2 x + 2\cos x \sin x}{\cos x + \sin x}$"
Please explain how can I get this to be to equal to $2\cos x $?
You've done fine so far. Just factor out $2\cos x$ from the numerator: $$2 \cos^2 x + 2 \cos x \sin x = 2\cos x(\cos x + \sin x)$$
With that as your numerator, simply cancel the common factor $(\cos x + \sin x)$ from numerator and denominator, leaving you with the desired $2\cos x$.