SOLUTION: Proving identities. How do I prove that: i) tanē(x) - sinē(x) = tanē(x) sinē(x) ii) (1+ cos(x) + cos(2x))/ (sin(x) + sin(2x)) = cot (x) I tried question 1 and i got stuck at

Algebra ->  Trigonometry-basics -> SOLUTION: Proving identities. How do I prove that: i) tanē(x) - sinē(x) = tanē(x) sinē(x) ii) (1+ cos(x) + cos(2x))/ (sin(x) + sin(2x)) = cot (x) I tried question 1 and i got stuck at      Log On


   

Question 1106380: Proving identities.
How do I prove that:
i) tanē(x) - sinē(x) = tanē(x) sinē(x)
ii) (1+ cos(x) + cos(2x))/ (sin(x) + sin(2x)) = cot (x)
I tried question 1 and i got stuck at sin2x - cos2x - cos4x which seemed quite wrong to me. Thanks :)
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
i) tanē(x) - sinē(x) = tanē(x) sinē(x) 

change tanē(x) to sinē(x)/cosē(x)

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

Get LCD of cosē(x)

%28sin%5E2%28x%29-sin%5E2%28x%29cos%5E2%28x%29%29%2Fcos%5E2%28x%29

Factor out sinē(x) on top:

%28sin%5E2%28x%29%281-cos%5E2%28x%29%29%29%2Fcos%5E2%28x%29

Replace 1-cosē(x) by sinē(x)

%28sin%5E2%28x%29%28sin%5E2%28x%29%29%29%2Fcos%5E2%28x%29

Break into a product:

%28sin%5E2%28x%29%5E%22%22%29%2A%28%28sin%5E2%28x%29%29%2Fcos%5E2%28x%29%29

Replace %28sin%5E2%28x%29%29%2Fcos%5E2%28x%29 by tan%5E2%28x%29

sin%5E2%28x%29%5E%22%22%2Atan%5E2%28x%29

Turn backward

tan%5E2%28x%29%5E%22%22%2Asin%5E2%28x%29

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

ii) %281%2B+cos%28x%29+%2B+cos%282x%29%29%2F+%28sin%28x%29+%2B+sin%282x%29%29%22%22=%22%22cot%28x%29

Use identities for cos(2x) and sin(2x)

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

%28cos%28x%29+%2B+2cos%5E2%28x%29%29%2F+%28sin%28x%29+%2B+2sin%28x%29cos%28x%29%29

Factor out common factors on top and bottom:





cos%28x%29%2Fsin%28x%29

cot%28x%29

Edwin