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There can be no significant simplicfication if some of the angles are theta and others are x, so assume that all angles are x. [csc(x) - cot(x)]*[cos(x) + 1] =[1/sin(x) - cos(x)/sin(x)]*[cos(x) + 1] =1/sin(x)*[1 - cos(x)]*[cos(x) + 1] =1/sin(x)*[1 - cos2(x)] =1/sin(x)*[sin2(x)] = sin(x)
Until an "equals" sign shows up somewhere in the expression, there's nothing to prove.
The derivative of (sin (theta))^.5 is (cos(theta))/(2sin(theta))
whats the big doubt,cot/tan+1= 1+1= 2
For a start, try converting everything to sines and cosines.