We seek:
#x=\cot \theta + \csc \theta + {1}/{\cot \theta - \csc \theta} + 1#
Plug in #\csc \theta={1}/{\sin \theta}#, #\cot \theta={\cos \theta}/{\sin \theta}#, then:
#x={\cos \theta}/{\sin \theta} + {1}/{\sin \theta} + {1}/{{\cos \theta}/{\sin \theta} - {1}/{\sin \theta}} + 1#
#x={\cos \theta+1}/{\sin \theta} + {\sin \theta}/{\cos - 1} + 1#
#x={\cos^2 \theta-1+\sin^2 \ theta}/{\sin \theta(\cos - 1)} + 1#
But #\sin^2 \theta + \cos^2 \theta =1# so the fraction cancels leaving just the constant #1#.