By forming and solving a quadratic equation, solve the equation 5*cosec(x) + cosec^2(x) = 2 - cot^2(x) in the interval 0<x<2*pi, giving the values of x in radians to three significant figures.

To solve the equation given in the question, we must first express the given equation in terms of only one trigonometric function, i.e. either all in cosec or all in cot. The easiest route would be cosec, since this will avoid square roots. To do this, we use the trigonometric identity 1+cot²(x)=cosec²(x) (obtained from sin²(x) + cos²(x) = 1 and dividing through by sin²(x)). We see from this identity that if we were to express 5*cosec(x) in terms of cot(x), we would obtain a square root term.

Then, by replacing the cot²(x) term with cosec²(x) - 1, we get:
5*cosec(x) + cosec²(x) = 2 - (cosec²(x) - 1).

Rearraging this, we obtain:
2cosec²(x) + 5*cosec(x) - 3 = 0 which is a quadratic equation.

We then solve this quadratic equation, either by factorising or using the quadratic-formula equation to get that:
cosec(x) = 1/2 or cosec(x) = -3

We know cosec(x)=1/sin(x), so we obtain:
sin(x)= 2 or sin(x)= -1/3

However, we know that -1<= sin(x) <=1, hence sin(x) = 2 is invalid, so sin(x) = -1/3.

Looking at the graph of sin(x), we can see that this yields two values of x in the interval (0,2*pi).

Then, we can use a calculator to get x=3.48, 5.94 to 3 significant figures.