Simplify (1+cos(x/2)-sin(x/2))/(1-cos(x/2)-sin(x/2))

Recall the double angle identities:

cos(x/2) = cos(x/4 + x/4) = cos²(x/4) - sin²(x/4)

sin(x/2) = sin(x/4 + x/4) = 2 sin(x/4) cos(x/4)

So

(1 + cos(x/2) - sin(x/2)) / (1 - cos(x/2) - sin(x/2))

= (1 + cos²(x/4) - sin²(x/4) - 2 sin(x/4) cos(x/4)) / (1 - cos²(x/4) + sin²(x/4) - 2 sin(x/4) cos(x/4))

sin²(x) + cos²(x) = 1, so

(1 + cos(x/2) - sin(x/2)) / (1 - cos(x/2) - sin(x/2))

= (2 cos²(x/4) - 2 sin(x/4) cos(x/4)) / (2 sin²(x/4) - 2 sin(x/4) cos(x/4))

= (cos(x/4) / sin(x/4)) • (2 cos(x/4) - 2 sin(x/4)) / (2 sin(x/4) - 2 cos(x/4))

=  (cos(x/4) / sin(x/4)) • (-1)

= - cot(x/4)

Step-by-step explanation:

In a right triangle, the hypotenuse is always the longest side. Since AC = BC, AB is the Hypotenuse.

Using Pythagoras theorem