Question

least positive value of csc(4 theta +3) sin( 5 theta - 6) =1

Answer 1

csc(4θ+3) sin(5θ-6) = 1
That's surely an odd sort of expression. Still, since cscθ = 1/sinθ, we have

sin(5θ-6) = sin(4θ+3)
So, we want
5θ-6 = 4θ+3
θ = 9
That seems kind of big, so what if we consider that sin(π-x) = sin(x). Then we have

5θ-6 = π-(4θ+3)
9θ = π+9
θ = 1 + π/9
Better, but we also know that
sin(π+x) = -sin(x), so
5θ-6 = -(π+(4θ+3))
9θ = 3-π
θ < 0 so we don't want that.

sin(2π-x) = -sin(x), so
5θ-6 = -(2π-(4θ+3))
θ = 9-2π

Hmmm. The problem is, that wolframalpha shows a solution at about θ=0.67 or roughly (1+π/9)/2. Maybe you can tweak the alternatives till you get that one.

The graph is at

http://www.wolframalpha.com/input/?i=solve+sin%285x-6%29+%3D+sin%284x%2B3%29+for+x+%3D+0..3