If sinX plus cosecx equals 2 find sin40x plus cosec50x?

sin x + csc x = 2

sin x + 1/sin x = 2

(sin2 x + 1)/sin x = 2/1 (cross multiply)

sin2 x + 1 = 2sin x (subtract 2sin x to both sides)

sin2 x - 2sinx + 1 = 0

(sin x - 1)2 = 0 (take the square root of both sides)

sin x - 1 = 0 (add 1 to both sides)

sin x = 1

x = sin-1 1 = 90⁰

40x = 40(90⁰) = 3,600⁰

sin 40x = sin (3,600⁰ ) = 0

50x = 50(90⁰) = 4500⁰

sin 50x = sin (4,500⁰) = 0, so that csc 50x is undefined, we cannot divide by 0.

Then, sin40x + csc50x is undefined.