How would I solve this on the interval?

Question

How would I solve this equation on the interval? I know to subtract something but after that I am lost.

4 csc 𝑥 + sin 𝑥 + 5 = 0

Answer 1

4/sinx + sinx + 5 = 0

multiply by sinx

4 + sin^2(x) + 5sinx = 0

sin^2(x) + 5sin(x) + 4 = 0

let y = sin(x)

y^2 + 5y +4 = 0

factor

(y+1)(y+4) = 0

set each factor = 0 and solve for y

y =-1 and y =-4

sinx =-1 x = 3pi/2 + 2npi where n= any integer

sinx=-4 is impossible as sines are only between -1 and 1

only solution is x =3pi/2 + 2npi

Answer 2