SOLUTION: sin(3x)/sin x= 2 cos(2x)+1 cot x/(csc x+1)= (csc x -1)/cot x

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Question 635009: sin(3x)/sin x= 2 cos(2x)+1



cot x/(csc x+1)= (csc x -1)/cot x






Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
sin%283x%29%2Fsin%28x%29 = 2cos%282x%29+%2B+1

We work with the left side

We write 3x as 2x+x

sin%282x%2Bx%29%2Fsin%28x%29 

Use the identity sin%28alpha%2Bbeta%29=sin%28alpha%29cos%28beta%29%2Bcos%28alpha%29sin%28beta%29
with alpha=2x and beta=x

+%28sin%282x%29cos%28x%29%2Bcos%282x%29sin%28x%29%29%0D%0A%0D%0A%2Fsin%28x%29

Break into the sum of two fractions with the same denominator
 
sin%282x%29cos%28x%29%2Fsin%28x%29 + cos%282x%29sin%28x%29%2Fsin%28x%29

Cancel the sin(x)'s in the second term

sin%282x%29cos%28x%29%2Fsin%28x%29 + cos%282x%29cross%28sin%28x%29%29%2Fcross%28sin%28x%29%29

sin%282x%29cos%28x%29%2Fsin%28x%29 + cos%282x%29

Use the identity sin%282theta%29=2sin%28theta%29cos%28theta%29 for sin%282x%29

2sin%28x%29cos%28x%29cos%28x%29%2Fsin%28x%29 + cos%282x%29

Cancel the sin(x)'s in the first term

2cross%28sin%28x%29%29cos%28x%29cos%28x%29%2Fcross%28sin%28x%29%29 + cos%282x%29

2cos%28x%29cos%28x%29 + cos%282x%29

2cos%5E2%28x%29 + cos%282x%29

Use the identity cos%282theta%29=2cos%5E2%28theta%29-1 solved for 2cos%5E2%28theta%29
or 2cos%5E2%28theta%29=cos%282theta%29%2B1

cos%282x%29%2B1 + cos%282x%29

Combine like terms:

2cos%282x%29+%2B+1

--------------------------

cot%28x%29%2F%28csc%28x%29%2B1%29 = %28csc%28x%29-1%29%2F%28cot%28x%29%29

We work with the left side:

Multiply the left sides by the conjugate of the denominator
over itself.  That is multiply by %28csc%28x%29-1%29%2F%28csc%28x%29-1%29

cot%28x%29%2F%28csc%28x%29%2B1%29·%28csc%28x%29-1%29%2F%28csc%28x%29-1%29

cot%28x%29%2F%28csc%28x%29%2B1%29%2A%28csc%28x%29-1%29%2F%28csc%28x%29-1%29

%28cot%28x%29%2A%28csc%28x%29-1%29%29%2F%28csc%5E2%28x%29-1%29

We use the identity 1%2Bcot%5E2%28theta%29=csc%5E2%28theta%29 to replace the
csc%5E2%28x%29 in the denominator.

%28cot%28x%29%2A%28csc%28x%29-1%29%29%2F%281%2Bcot%5E2%28x%29-1%29

%28cot%28x%29%2A%28csc%28x%29-1%29%29%2Fcot%5E2%28x%29

%28cross%28cot%28x%29%29%2A%28csc%28x%29-1%29%29%2Fcot%5Ecross%282%29%28x%29

%28csc%28x%29-1%29%2F%28cot%28x%29%29

Edwin