Expert: Ahmed Salami - 10/11/2009

Question
find the limit as theta goes to zero of the function cos theta minus 1 divided by sin theta.

Answer
Hi Clay
(cosθ - 1)/sinθ = (cosθ - 1)(cosθ + 1)/sinθ.(cosθ + 1)
Note that we have multiplied both top and bottom by the same expression (cosθ + 1) and so that preserves the initial expression i.e it still has the same value.
So,
(cosθ - 1)/sinθ = (cos²θ - 1)/sinθ.(cosθ + 1)
               = -sin²θ/sinθ.(cosθ + 1)
Remember that cos²θ + sin²θ = 1
(cosθ - 1)/sinθ = -sinθ/(cosθ + 1)
Therefore,
lim (cosθ - 1)/sinθ = lim [-sinθ/(cosθ + 1)]
x→0                   x→0
                   = -sin(0)/[cos(0) + 1]
                   = 0/(0+1)
                   = 0/1
                   = 0

Regards