Substitute #cos^2(theta)/sin^2(theta)# for #cot^2(theta)# and #1/sin(theta)# for #csc(theta)#:
#cos^2(theta)/sin^2(theta) = 1/sin(theta) + 1#
Multiply both sides by #sin^2(theta)#:
#cos^2(theta) = sin(theta) + sin^2(theta)#
Substitute #1 - sin^2(theta)# for #cos^2(theta)#
#1 - sin^2(theta) = sin(theta) + sin^2(theta)#
Combine like terms:
#2sin^2(theta) + sin(theta) - 1 = 0#
Factor
#(2sin(theta) - 1)(sin(theta) + 1) = 0#
Set each factor equal to zero:
#(2sin(theta) - 1) = 0 and (sin(theta) + 1) = 0#
Solve both equations for #sin(theta)#;
#sin(theta) = 1/2 and sin(theta) = -1#
The left equation happens twice; the right one only once:
#theta = sin^-1(1/2), theta = 180 - sin^-1(1/2), and theta = sin^-1(-1)#
#theta = 30^@, theta = 150^@, and theta = 270^@#
