Evaluate cot x, given that $\displaystyle cot x=sin x$?
cot(x) = sin(x)
cos(x) / sin(x) = sin(x)
cos(x) = sin^2(x)
cos(x) = 1 -cos^2(x)
cos^2(x) +cos(x) -1 = 0
By the Quadratic Formula,
cos(x) = {-1 +,-sqrt[1^2 -4(1)(-1)]} / 2(1)
cos(x) = {-1 +,-sqrt(5)}/2
When cos(x) = (1/2)[-1 +sqrt(5)] = 0.618 ----umm, golden ratio conjugate.
angle x then is in the 1st and 4th quadrants.
adjacent side = 0.618
hypotenuse = 1
opposite side = sqrt[1^2 -(0.618)^2] = +,-0.786178
So, cot(x) = adj/opp = 0.618 / (+,-0.786178) = +,-0.786 -------answer.
When cos(x) = (1/2)[-1 -sqrt(5)] = -1.618
Cannot be. The minimum value of the cosine of any angle ios only (-1).
So, reject this factor.