The quotient properties of tangent and cotangent are: tanx = sinx ÷ cosx and cotx = cosx ÷ sinx. why are these true based on the unit circle?

Hope this helps:

Given The unit circle, a radius can be drawn from the origin to any point on the circle in the direction of theta relative to the x axis. This creates the coordinate (x,y)

Drawing a right triangle from this gives you a horizontal length of x, vertical length of y and a hypotenuse of 1.

Use SOHCAHTOA to find tan (theta) and cot (theta):

sin(theta) = y/1 = y

cos(theta) = x/1 = x

tan(theta) = O/A = y/x = sin(theta) / cos(theta)

cot(theta) = A/O = x/y = cos(theta) / sin(theta)