Apply the identities cottheta = costheta/sinthetacotθ=cosθsinθ and csctheta = 1/sinthetacscθ=1sinθ:
cos^2theta/sin^2theta + 1/sintheta = 1cos2θsin2θ+1sinθ=1
cos^2theta/sin^2theta + sintheta/sin^2theta = 1cos2θsin2θ+sinθsin2θ=1
(cos^2theta + sin theta)/sin^2theta = 1cos2θ+sinθsin2θ=1
cos^2theta + sin theta = sin^2thetacos2θ+sinθ=sin2θ
Apply the identity sin^2theta + cos^2theta = 1 ->cos^2theta = 1 - sin^2thetasin2θ+cos2θ=1→cos2θ=1−sin2θ
1 - sin^2theta + sin theta - sin^2theta = 01−sin2θ+sinθ−sin2θ=0
-2sin^2theta + sin theta + 1 = 0−2sin2θ+sinθ+1=0
-2sin^2theta + 2sintheta - sin theta + 1 = 0−2sin2θ+2sinθ−sinθ+1=0
-2sintheta(sin theta - 1) - 1(sin theta - 1) = 0−2sinθ(sinθ−1)−1(sinθ−1)=0
(-2sintheta - 1)(sin theta - 1) = 0(−2sinθ−1)(sinθ−1)=0
sintheta = -1/2 and sin theta = 1sinθ=−12andsinθ=1
theta = (7pi)/6, (11pi)/6 and pi/2θ=7π6,11π6andπ2
However, since pi/2π2 renders the equation undefined, that solution is extraneous. Hence, our solution set is {(7pi)/6, (11pi)/6}{7π6,11π6}.
Hopefully this helps!
