Trigo fun.. :D

(Sin x + Cos x) exceeds Sqrt (2) means x are complex numbers. Take easy range for Sin x and Cos x for a simple x,

Let a = Sin x, Cox = Sqrt (1 - a^2)

a + Sqrt (1 - a^2) = 2

=> 2 a^2 - 4 a + 3 = 0

Now, we want Sin 2 x and it is still 2 Sin x Cox,

2 a Sqrt (1 - a^2 ) = 2 a (2 - a) = 4 a - 2 a^2 = 4 a - 4 a + 3 = 3 {From above}

$\sin 2x = 2\sin x \cos x$

$(\sin x + \cos x)^2 = 4$

$\sin^2 x + 2\sin x\cos x + \cos^2 x = 4 \implies \boxed{\sin^2 x + \cos^2 x = 1}$

$2\sin x\cos x = 3 \implies \sin 2x = \boxed{3}$