SOLUTION: please help prove the identity cos(u-v)/cos(u)sin(v)= tan(u)+cot(v) thank you!!!

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Question 240783: please help prove the identity
cos(u-v)/cos(u)sin(v)= tan(u)+cot(v)
thank you!!!
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
cos%28u-v%29%2F%28cos%28u%29sin%28v%29%29=+tan%28u%29%2Bcot%28v%29
Since there is no (u-v) on the right side, we will use cos%28A-B%29+=+cos%28A%29cos%28B%29+%2B+sin%28A%29sin%28B%29 on the cos(u-v) giving us:

Since there are two terms on the right side, we will split the fraction on the left into two terms. (Think of it as "unadding".):

As you can see, we can do some canceling in each fraction:

leaving:
cos%28v%29%2Fsin%28v%29+%2B+sin%28u%29%2Fcos%28u%29=+tan%28u%29%2Bcot%28v%29
Since cos(v)/sin(v) = cot(v) and sin(u)/cos(u) = tan(u) we can substitute and get:
cot%28v%29+%2B+tan%28u%29+=+tan%28u%29%2Bcot%28v%29
And, using the Commutative Property of Addition, we can change the order on the left to:
tan%28u%29%2B+cot%28v%29+=+tan%28u%29%2Bcot%28v%29
And we are done.