If tan^3(theta) -1/tan(theta)-1 - sec^2(theta) + 1 = 0 find cot(theta)

Recall that , so we can write everything in terms of :

Let , so that

With some rewriting we get

Clearly we cannot have , or .

The numerator determines when the expression on the left reduces to 0:

Completing the square gives

so that

The second equation gives no real-valued solutions because squaring any real number gives a positive real number. (I'm assuming we don't care about complex solutions.) So we're left with only

which again gives two cases,

Then when , we can find by taking the reciprocal, so we get