Question

What is the complete solution set of the equation 1+2cosec x=(-sec²x/2)/2?

Answer 1

`x = 2kpi- pi/2 = (4k - 1 )pi/2, k = 0, +-1, +-2, +-3, ...`.

This angle separates `Q_3 and Q_4`.

Explanation:

With `t = tan (x/2),`

`sin x = ( 2 t )/( 1 + t^2) and sec^2 ( x/2 ) = 1 + t^2`.

Now, the given equation changes to

`1 + 2 ( (1 + t^2 )/(2 t )) +1/2( 1 + t^2 ) = 0`, giving

`t^3 + 2t^2 + 3t + 2 = ( t + 1 )( t^2 + t + 2 ) = 0`. Solving,

`t = -1 rArr tan ( x/2 ) = - 1 = tan ( -pi/4 )` that gives solution in

multitude as

`x/2 = kpi - pi/4 rArr x = 2kpi- pi/2 = (4k - 1 )pi/2`,

`k = 0, +-1, +-2, +-3, ...`.

Also,

#t^2 +t + 2 = 0, giving t = complex ( unreal ) values, and so,

there are no other solutions.

Graphical depiction, in t-intercept t = -1, for unique solution of

`t^3 + 2t^2 + 3t + 2 = 0` .
graph{y-x^3-2x^2-3x-2=0}