SOLUTION: Establish the identity (Prove two sides are equal). (cot x + csc x -1)/(cot x - csc x +1) = csc x + cot x

Algebra ->  Trigonometry-basics -> SOLUTION: Establish the identity (Prove two sides are equal). (cot x + csc x -1)/(cot x - csc x +1) = csc x + cot x      Log On


   

Question 1205700: Establish the identity (Prove two sides are equal).
(cot x + csc x -1)/(cot x - csc x +1) = csc x + cot x
Answer by Edwin McCravy(20063) About Me  (Show Source):
You can put this solution on YOUR website!




Replace the trig ratios cos(x) and sin(x) by their initials C and S:

%28C%2FS+%2B+1%2FS+-1%29%2F%28C%2FS+-+1%2FS+%2B1%29

%28C+%2B+1+-S%29%2F%28C+-+1+%2BS%29

%28%28C-S%29%2B1%29%2F%28%28C%2BS%29-1%29

%28%28C-S%29%2B1%29%2F%28%28C%2BS%29-1%29%22%22%2A%22%22%28%28C%2BS%29%2B1%29%2F%28%28C%2BS%29%2B1%29

%28++%28C-S%29+%28C%2BS%29%2B1%28C-S%29%2B1%28C%2BS%29%2B1%29%2F%28%28C%2BS%29%5E2-1%29

%28C%5E2-S%5E2%2BC-S%2BC%2BS%2B1%29%2F%28C%5E2%2B2CS%2BS%5E2-1%29

%28C%5E2-S%5E2%2B2C%2B1%29%2F%28%28C%5E2%2BS%5E2%29%2B2CS-1%29

Since cos%5E2%28x%29%2Bsin%5E2%28x%29=1, Replace the 1 in the top by
C%5E2%2BS%5E2 and vice versa in the bottom:

%28C%5E2-S%5E2%2B2C%2BC%5E2%2BS%5E2%29%2F%281%2B2CS-1%29

%282C%5E2%2B2C%29%2F%282CS%5E%22%22%29

%282C%5E2%29%2F%282CS%5E%22%22%29%2B%282C%5E%22%22%29%2F%282CS%5E%22%22%29

C%2FS%2B1%2FS

cos%28x%29%2Fsin%28x%29%2B1%2Fsin%28x%29

cot%28x%29%2Bcsc%28x%29

csc%28x%29%2Bcot%28x%29

Whew!

Edwin