if Cot x + cosec x=2 sin x then number of solution of equation in [0,3π) is _______

Question

Jee question 

Answer 1

The easiest way to answer this question is to graph the equation which shows that there are 4 solutions in the half open interval.

 

To get them algebraically, write the equation as:

(cos x/sin x) + (1/sin x) = 2 sin x

 

cos x + 1 = 2 sin² x = 2(1 - cos² x)

divide through by  1 + cos x: 1 = 2(1 - cos x) then collect and simplify to get

cos x = 1/2 which gives roots at 60°, 300°, and  420°

but 1+ cos x = 0 also provides a solution at 180°