SOLUTION: Determine the constants a and b such that (-3 + 4(cos^2 x))/(1 - 2(sin x)) = a + b(sin x) for all values of x

The numerator is


    -3 + 4cos^2(x) = -3 + 4*(1-sin^2(x)) = 1 - 4sin^2(x) = (1-2sin(x))*(1+2sin(x)).



Now,   = 


             (after canceling the factor (1-2sin(x)) in the numerator and denominator)


     = 1 + 2sin(x).


Therefore,  in this identity  a= 1,  b= 2.



Surely, the identity is valid only over the domain, which is  the entire number line excluding the roots of the denominator


    1 - 2sin(x) = 0,    i.e.  except   x= arcsin(1/2) = .         ANSWER