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Question 151205: sin(x+y)-2sinx+sin(x-y)/cos(x+y)-2cosx+cos(x-y)= tan x: sin(x+y)-2sinx+sin(x-y)/cos(x+y)-2cosx+cos(x-y)= tan x
Answer by Edwin McCravy(2087) About Me  (Show Source):
You can put this solution on YOUR website!
(sin(x+y)-2sinx+sin(x-y))/(cos(x+y)-2cosx+cos(x-y))= tan(x)

We need these four identities to use on the left side:

sin(x+y)=sin(x)cos(y)+cos(x)sin(y)
sin(x-y)=sin(x)cos(y)-cos(x)sin(y)
cos(x+y)=cos(x)cos(y)-sin(x)sin(y)
cos(x-y)=cos(x)cos(y)+sin(x)sin(y)

(sin(x)cos(y)+cos(x)sin(y)-2sin(x)+sin(x)cos(y)-cos(x)sin(y))/(cos(x)cos(y)-sin(x)sin(y)-2cos(x)+cos(x)cos(y)+sin(x)sin(y))= tan(x)

Cancel positive and negative like terms
in numerator and in denominator:

(sin(x)cos(y)+cross(cos(x)sin(y))-2sin(x)+sin(x)cos(y)-cross(cos(x)sin(y)))/(cos(x)cos(y)-cross(sin(x)sin(y))-2cos(x)+cos(x)cos(y)+cross(sin(x)sin(y)))= tan(x)

(sin(x)cos(y)-2sin(x)+sin(x)cos(y))/(cos(x)cos(y)-2cos(x)+cos(x)cos(y))= tan(x)

Combine like terms:

(2sin(x)cos(y)-2sin(x))/(2cos(x)cos(y)-2cos(x))= tan(x)

Factor out 2sin(x) in the top and 2cos(x) in the
bottom:

2sin(x)(cos(y)-1)/(2cos(x)(cos(y)-1))= tan(x)

Cancel the 2's and the (cos(y)-1)'s.

cross(2)sin(x)cross((cos(y)-1))/(cross(2)cos(x)cross((cos(y)-1)))= tan(x)

sin(x)/cos(x)=tan(x)

Then use identity sin(x)/cos(x)=tan(x)

tan(x)=tan(x)

Edwin