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!
