SOLUTION: 22. verify the identity csc(-x)/sec(-x)=-cot x

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Question 169978: 22. verify the identity csc(-x)/sec(-x)=-cot x
Found 2 solutions by jim_thompson5910, jojo14344:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Note: I'm ONLY manipulating the left side. I'm showing the right side for comparison


csc%28-x%29%2Fsec%28-x%29=-cot%28x%29 Start with the given equation


%281%2Fsin%28-x%29%29%2F%281%2Fcos%28-x%29%29=-cot%28x%29 Use the identities csc%28A%29=1%2Fsin%28A%29 and sec%28A%29=1%2Fcos%28A%29 to get the left side in terms of sine and cosine


%281%2Fsin%28-x%29%29%2A%28cos%28-x%29%2F1%29=-cot%28x%29 Multiply the first fraction by the reciprocal of the second fraction.


cos%28-x%29%2Fsin%28-x%29=-cot%28x%29 Combine and multiply the fractions.


cos%28x%29%2Fsin%28-x%29=-cot%28x%29 Use the identity cos%28-A%29=cos%28A%29


cos%28x%29%2F%28-sin%28x%29%29=-cot%28x%29 Use the identity sin%28-A%29=-sin%28A%29


-cos%28x%29%2Fsin%28x%29=-cot%28x%29 Reduce


-cot%28x%29=-cot%28x%29 Use the identity cos%28A%29%2Fsin%28A%29=cot%28A%29

Answer by jojo14344(1513) About Me  (Show Source):
You can put this solution on YOUR website!

Re-writing: csc%28-x%29%2Fsec%28-x%29=-cot%28x%29
Via Reciprocal Identites: csc%28x%29=1%2Fsin%28x%29; sec%28x%29=1%2Fcos%28x%29
Subst. & becomes:
%281%2Fsin%28-x%29%29%2F%281%2Fcos%28-x%29%29=-cot%28x%29
cos%28-x%29%2Fsin%28-x%29=-cot%28x%29
Via Quotient Identities: cot%28x%29=cos%28x%29%2Fsin%28x%29
Subst. & becomes:
cot%28-x%29=-cot%28x%29
highlight%28-cot%28x%29=-cot%28x%29%29
Thank you,
Jojo