Expert: Steve Holleran - 12/6/2007

Question
Hi, I need help with this question:

Prove the following identity: (cos^2(u-v))/(cos u sin v)=(tan u+cot v)/(sec(u-v))

Answer
Hello Emily,

I would start on the right side:

(tan u + cot v) / sec(u-v)

= [sin u/cos u + cos v/sin v] / [1/cos(u-v)]

Now multiply the top and bottom by cos(u-v)

= ([sin u/cos u + cos v/sin v] * cos(u-v)) /
                 [1/cos(u-v) * cos(u-v)]

On the top, combine the bracket expressions over a common denominator of cos u * sin v:

= ([sin u sin v + cos v cos u]/[cos u sin v] * cos(u-v)

and the bottom just becomes 1: 1/cos(u-v) * cos(u-v) = 1

So now if you look at the big bracket on the top, the numerator is the identity for cos(u-v):

[cos(u-v) / cos u sin v] * cos(u-v)  / 1

= cos^2 (u-v) / cos u sin v

Hope you could follow okay.

Steve