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Answer 1 · 2017-12-04T14:56:19

$x = 1.571, 3.309, 6.116$ $x=1.571,3.309,6.116$

${csc}^{2} x + 5 csc x - 6 = 0$ $csc^2x+5cscx-6=0$ can be factorized as under

${csc}^{2} x + 6 csc x - csc x - 6 = 0$ $csc^2x+6cscx-cscx-6=0$

or $csc x (csc x + 6) - 1 (csc x + 6) = 0$ $cscx(cscx+6)-1(cscx+6)=0$

or $(csc x - 1) (csc x + 6) = 0$ $(cscx-1)(cscx+6)=0$

i.e. either $csc x - 1 = 0$ $cscx-1=0$ i.e. $csc x = 1$ $cscx=1$ and $x = \frac{π}{2} = 1.571$ $x=pi/2=1.571$

or $csc x + 6 = 0$ $cscx+6=0$

i.e. $csc x = - 6$ $cscx=-6$ i.e. $x = 6.116$ $x=6.116$ or $3.309$ $3.309$ (in radians)