SOLUTION: Verify the identity sin(x)/(1+cos(x))+(1+cos(x))/sin(x)=2csc(x)

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Question 249656: Verify the identity
sin(x)/(1+cos(x))+(1+cos(x))/sin(x)=2csc(x)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
sin%28x%29%2F%281%2Bcos%28x%29%29%2B%281%2Bcos%28x%29%29%2Fsin%28x%29=2csc%28x%29

Since the right side has 1 term and the left side has two terms, we will start by adding the terms on the left. Of course to add fractions we must have common denominators:

which simplifies:


Now we can add:

Since sin%28x%29%5E2+%2B+cos%28x%29%5E2+=+1 we get:
%281+%2B+1%2B2cos%28x%29%29%2F%28%281%2Bcos%28x%29%29%28sin%28x%29%29%29+=+2csc%28x%29
%282+%2B+2cos%28x%29%29%2F%28%281%2Bcos%28x%29%29%28sin%28x%29%29%29+=+2csc%28x%29
We now have a simplified, one-term expression on the left. Since the right side has a factor of 2, we'll factor out a 2 on the left:
%282%281+%2B+cos%28x%29%29%29%2F%28%281%2Bcos%28x%29%29%28sin%28x%29%29%29+=+2csc%28x%29
As you can see, the (1 + cos(x))'s cancel leaving:
2%2Fsin%28x%29+=+2csc%28x%29
or
2%281%2Fsin%28x%29%29+=+2csc%28x%29
Since 1/sin(x) = csc(x) we have:
2csc%28x%29+=+2csc%28x%29
And we are finished.