How do I factor: cotx^2+4cscx+5 ?

Question

The whole question is:
Solve the equation for solutions over the interval [0, 2pi)

I know factoring is the first step.

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Answer 1 · 2018-04-29T00:54:53

#(cscx+2)^2#

#cotx^2+4cscx+5=0#

Apply modified pythagorean identity: #cot^2x=csc^2x-1#
#(csc^2x-1)+4cscx+5=0#

#csc^2x+4cscx+4=0#

#(cscx+2)(cscx+2)=0#

#cscx=-2#
#sinx=-1/2#

#x= (7pi)/6+2pin#
#x= (11pi)/6+2pin#

graph{(cotx)^2+4cscx+5 [-10, 10, -5, 5]}