Expert: Steve Holleran - 5/7/2008

Question
Verify the identity:  1/1+cost=(csct-cott)csct

Answer
Hi Robert,

Okay, so we've got

                     1 / (1 + cos t) = (csc t - cot t) * csc t

I would work the right side, changing all the expressions back in terms of sines and cosines:

Right:  (csc t - cot t)* csc t = (1/sin t - cos t / sin t) * 1/sin t

          = ( (1 - cos t)/ sin t) * 1/sin t

          = (1 - cos t) / sin^2 t

          = (1 - cos t) / (1 - cos^2 t)     Pythagorean identity

          = (1 - cos t) / [(1 - cos t)(1 + cos t)]

Now cancel the 1 - cos t factors and you have it.

Steve