Solve the trigonometric equations exactly on the interval 0

Solve the trigonometric equations exactly on the interval 0 x<2

sinx + cscx = 2

sin(2x) = 3 sinx

sin(3x) cos(2x) - cos(3x) sin(2x) = 1

Please show and explain work on the first one i got to sin²x -2sinx + 1 = 0 then i didn\'t know what to do.

Solution

sinx + cscx=2

sinx + 1/sinx=2

sin²x+1=2sinx

sin²x-2sinx+1=0

Let sinx =t

t^2-2t+1=0

solving by factoring

(t-1)(t-1)=0

Therefore

t-1=0

t=1

and sinx =t

Therefore sinx =1

taking sin ^-1 on both sides

sin^-1(sin x)=sin^-11

x=90