Consider the following identities:
•#csctheta = 1/sintheta#
•#cottheta = 1/(tan theta) = 1/(sintheta/costheta) = costheta/sintheta#
Applying both these identities to the initial equation, we have that:
#1/sinx + cosx/sinx = 1#
#(1 + cosx)/sinx = 1#
#1 + cosx = sinx#
#(1 + cosx)^2 = (sinx)^2#
#1 + 2cosx + cos^2x = sin^2x#
#cos^2x - sin^2x + 2cosx + 1 = 0#
Converting the #sin^2x# to #1 - cos^2x# in accordance with the pythagorean identity #sin^2x + cos^2x = 1#:
#cos^2x - (1 - cos^2x) + 2cosx + 1 = 0#
#cos^2x - 1 + cos^2x + 2cosx + 1 = 0#
#2cos^2x + 2cosx = 0#
#2cosx(cosx + 1) = 0#
#cosx = 0 and cosx = -1#
#x = pi/2, (3pi)/2, pi#
However, #pi# is extraneous, since it makes the denominator equal to zero and therefore the expression undefined. The #(3pi)/2# is also extraneous, because it doesn't work in the initial equation.
Hence, the solution set is #{pi/2}#.
Hopefully this helps!
