SOLUTION: rewrite in terms of csc (1/(1+cosx))+(cosx)/(1-cosx)

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Question 635241: rewrite in terms of csc
(1/(1+cosx))+(cosx)/(1-cosx)
Found 2 solutions by Edwin McCravy, AnlytcPhil:
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
1%2F%281%2Bcos%28x%29%29 + cos%28x%29%2F%281-cos%28x%29%29

Since csc(x) = 1%2Fsin%28x%29 we'll get it first in terms of sin(x)

Get an LCD of %281+%2B+cos%28x%29%29%281-cos%28x%29%29

1%2F%281%2Bcos%28x%29%29·%281+-+cos%28x%29%29%2F%281-cos%28x%29%29 + cos%28x%29%2F%281-cos%28x%29%29·%281+%2B+cos%28x%29%29%2F%281%2Bcos%28x%29%29



%281+-+cos%28x%29%29%2F%281-cos%5E2%28x%29%29 + %28cos%28x%29%281+%2B+cos%28x%29%29%29%2F%281-cos%5E2%28x%29%29

Since the denominators 1-cosē(x) are equal to sinē(x)

%281+-+cos%28x%29%29%2Fsin%5E2%28x%29 + %28cos%28x%29%281+%2B+cos%28x%29%29%29%2Fsin%5E2%28x%29

Multiply the numerator of the second fraction out:

%281+-+cos%28x%29%29%2Fsin%5E2%28x%29 + %28cos%28x%29+%2B+cos%5E2%28x%29%29%2Fsin%5E2%28x%29

Combine the numerators over the common denominator

%281-cos%28x%29%2Bcos%28x%29+%2B+cos%5E2%28x%29%29%2Fsin%5E2%28x%29

%281%2B+cos%5E2%28x%29%29%2Fsin%5E2%28x%29

Replace cosē(x) by 1-sinē(x)

%281%2B+%281-sin%5E2%28x%29%29%29%2Fsin%5E2%28x%29

%281%2B+1-sin%5E2%28x%29%29%2Fsin%5E2%28x%29

%282-sin%5E2%28x%29%29%2Fsin%5E2%28x%29

Break into two fractions:

2%2Fsin%5E2%28x%29 - sin%5E2%28x%29%2Fsin%5E2%28x%29

2%2Fsin%5E2%28x%29 - cross%28sin%5E2%28x%29%29%2Fcross%28sin%5E2%28x%29%29

2%2Fsin%5E2%28x%29 - 1

Bring sinē(x) to the numerator as cscē(x)

2cscē(x) - 1

Edwin

Answer by AnlytcPhil(1806) About Me  (Show Source):
You can put this solution on YOUR website!
1%2F%281%2Bcos%28x%29%29 + cos%28x%29%2F%281-cos%28x%29%29

Since csc(x) = 1%2Fsin%28x%29 we'll get it first in terms of sin(x)

Get an LCD of %281+%2B+cos%28x%29%29%281-cos%28x%29%29

1%2F%281%2Bcos%28x%29%29·%281+-+cos%28x%29%29%2F%281-cos%28x%29%29 + cos%28x%29%2F%281-cos%28x%29%29·%281+%2B+cos%28x%29%29%2F%281%2Bcos%28x%29%29



%281+-+cos%28x%29%29%2F%281-cos%5E2%28x%29%29 + %28cos%28x%29%281+%2B+cos%28x%29%29%29%2F%281-cos%5E2%28x%29%29

Since the denominators 1-cosē(x) are equal to sinē(x)

%281+-+cos%28x%29%29%2Fsin%5E2%28x%29 + %28cos%28x%29%281+%2B+cos%28x%29%29%29%2Fsin%5E2%28x%29

Multiply the numerator of the second fraction out:

%281+-+cos%28x%29%29%2Fsin%5E2%28x%29 + %28cos%28x%29+%2B+cos%5E2%28x%29%29%2Fsin%5E2%28x%29

Combine the numerators over the common denominator

%281-cos%28x%29%2Bcos%28x%29+%2B+cos%5E2%28x%29%29%2Fsin%5E2%28x%29

%281%2B+cos%5E2%28x%29%29%2Fsin%5E2%28x%29

Replace cosē(x) by 1-sinē(x)

%281%2B+%281-sin%5E2%28x%29%29%29%2Fsin%5E2%28x%29

%281%2B+1-sin%5E2%28x%29%29%2Fsin%5E2%28x%29

%282-sin%5E2%28x%29%29%2Fsin%5E2%28x%29

Break into two fractions:

2%2Fsin%5E2%28x%29 - sin%5E2%28x%29%2Fsin%5E2%28x%29

2%2Fsin%5E2%28x%29 - cross%28sin%5E2%28x%29%29%2Fcross%28sin%5E2%28x%29%29

2%2Fsin%5E2%28x%29 - 1

Bring sinē(x) to the numerator as cscē(x)

2cscē(x) - 1

Edwin