Solve $\sin x + \cos x = \sqrt{1+k}$ for $\sin 2x$, $\sin x-\cos x$, and $\tan x$ in terms of $k$

Try this for the second one,

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

For the third one, solve the equations below for $\sin x$ and $\cos x$,

$$ \sin x + \cos x = \sqrt{1+k}$$ $$ \sin x- \cos x = \sqrt{1-k}$$

and then plug them into

$$\tan x= \frac{\sin x}{\cos x}$$