SOLUTION: Simplify: [(cscx-cotx)(cscx+cotx)]/cscx

Algebra ->  Trigonometry-basics -> SOLUTION: Simplify: [(cscx-cotx)(cscx+cotx)]/cscx      Log On


   

Question 156515: Simplify:
[(cscx-cotx)(cscx+cotx)]/cscx
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
Simplify:
%28%28csc%28x%29-cot%28x%29%29%28csc%28x%29%2Bcot%28x%29%29%29%2Fcsc%28x%29

FOIL out the top:

%28csc%5E2x%2Bcsc%28x%29cot%28x%29-cot%28x%29csc%28x%29-cot%5E2x%29%2Fcsc%28x%29

%28csc%5E2x%2Bcross%28csc%28x%29cot%28x%29-cot%28x%29csc%28x%29%29-cot%5E2x%29%2Fcsc%28x%29

%28csc%5E2x-cot%5E2x%29%2Fcsc%28x%29

use the Pythagorean identity 1%2Bcot%5E2alpha=csc%5E2alpha which
can be rearranged as csc%5E2alpha-cot%5E2alpha=1 to replace
the numerator by just 1.

1%2Fcsc%28x%29

Then use the reciprocal identity csc%28alpha%29=1%2Fsin%28alpha%29
to replace the denominator:

1%2F%281%2Fsin%28x%29%29

1÷1%2F%28sin%28x%29%29

Invert the second fraction and change division to
multiplication:

1×%28sin%28x%29%29%2F1

sin%28x%29

Edwin