SOLUTION: prove that -sin(x)-cos(x)cot(x)=-csc(x) is an identity

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Question 1201965: prove that -sin(x)-cos(x)cot(x)=-csc(x) is an identity
Found 2 solutions by mananth, Theo:
Answer by mananth(16946) About Me  (Show Source):
You can put this solution on YOUR website!
prove that -sin(x)-cos(x)cot(x)=-csc(x) is an identity
-sin(x)-cos(x)cot(x)
-sin%28x%29-%28cos%28x%29cos%28x%29%29%2Fsin%28x%29
LCM
%28-sin%5E2%28x%29-cos%5E2%28x%29%29%2Fsin%28x%29
%28-sin%5E2%28x%29-+%281-sin%5E2%28x%29%29%29%2Fsin%28x%29%29
%28+-sin%5E2%28x%29-+1%2Bsin%5E2%28x%29%29%2Fsin%28x%29....sin^2(x) cancel off
%28-1%2Fsin%28x%29%29
-cosec(x)
PROVED






Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
equation is:
-sin(x) - cos(x) * cot(x) = -csc(x)
csc(x) = 1/sin(x)
cot(x) = cos(x) / sin(x)
equation becomes:
-sin(x) - cos(x) * cos(x) / sin(x) = -1/sin(x)
multiply both sides of the equation by sin(x) to get:
-sin^2(x) - cos^2(x) = -1
factor the left side of the equation by -1 to get:
-1 * (sin^2(x) + cos^2(x) = -1
since sin^2(x) + cos^(x) = 1, you get:
-1 = -1.
this confirms the identity is correct.