(a) Where the expressions are defined, \[\frac{1+\sin x}{1-\sin x}-\frac{1-\sin x}{1+\sin x}=\frac{(1+\sin x)^2-(1-\sin x)^2}{1-\sin^2x}\] \[=\frac{4\sin x}{\cos^2x}=\frac{4\tan x}{\cos x},\] proving the identity.

(b) The endpoint \(x=\pi/2\) is undefined. For the remaining interval, multiply by \(\cos x\): \[4\tan x=8\tan x\cos x,\qquad4\tan x(1-2\cos x)=0.\] Therefore either \(\tan x=0\), giving \(x=0\), or \(\cos x=1/2\), giving \(x=\pi/3\). Thus \[\boxed{x=0,\quad\pi/3}.\]