Find real x such that `(3+2i sin x)/(1-2i sin x)` is purely real.

Let A={theta in (-pi /2,pi):(3+2i sin theta )/(1-2 sin theta ) is purely imaginary } Then the sum of the elements in A is

Prove that z=i^i, where i=sqrt-1, is purely real.

If S is the set of all real x such that (2x-1)/(2x^3+3x^2+x) is positive

Find the integrals of the functions sin x sin 2x sin 3x

find all the possible triplets (a_(1), a_(2), a_(3)) such that a_(1)+a_(2) cos (2x)+a_(3) sin^(2) (x)=0 for all real x.

The number of distinct real roots of the equation sin^(3)x +sin^(2)x sin x-sin x- sin 2x-2cos x=0 belonging to the interval (-(pi)/(2),(pi)/(2))

Find the range of the function f(x)=(1)/(2+sin3x) .

If z=x+i ya n dw=(1-i z)//(z-i) , then show that |w|=1 z is purely real.

Find the value of a for which the equation a sin(x+pi/4)=sin2x+9 will have real solution.