SOLUTION: sin(x+y)-2sinx+sin(x-y)/cos(x+y)-2cosx+cos(x-y)= tan x

Algebra ->  Algebra  -> Trigonometry-basics -> SOLUTION: sin(x+y)-2sinx+sin(x-y)/cos(x+y)-2cosx+cos(x-y)= tan x      Log On

Ad: Algebra Solved!™: algebra software solves algebra homework problems with step-by-step help!
Ad: Algebrator™ solves your algebra problems and provides step-by-step explanations!

   

Question 151205: sin(x+y)-2sinx+sin(x-y)/cos(x+y)-2cosx+cos(x-y)= tan x
Answer by Edwin McCravy(6938) About Me  (Show Source):
You can put this solution on YOUR website!
%28sin%28x%2By%29-2sinx%2Bsin%28x-y%29%29%2F%28cos%28x%2By%29-2cosx%2Bcos%28x-y%29%29=+tan%28x%29

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

sin%28x%2By%29=sin%28x%29cos%28y%29%2Bcos%28x%29sin%28y%29
sin%28x-y%29=sin%28x%29cos%28y%29-cos%28x%29sin%28y%29
cos%28x%2By%29=cos%28x%29cos%28y%29-sin%28x%29sin%28y%29
cos%28x-y%29=cos%28x%29cos%28y%29%2Bsin%28x%29sin%28y%29

%28sin%28x%29cos%28y%29%2Bcos%28x%29sin%28y%29-2sin%28x%29%2Bsin%28x%29cos%28y%29-cos%28x%29sin%28y%29%29%2F%28cos%28x%29cos%28y%29-sin%28x%29sin%28y%29-2cos%28x%29%2Bcos%28x%29cos%28y%29%2Bsin%28x%29sin%28y%29%29=+tan%28x%29

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

%28sin%28x%29cos%28y%29%2Bcross%28cos%28x%29sin%28y%29%29-2sin%28x%29%2Bsin%28x%29cos%28y%29-cross%28cos%28x%29sin%28y%29%29%29%2F%28cos%28x%29cos%28y%29-cross%28sin%28x%29sin%28y%29%29-2cos%28x%29%2Bcos%28x%29cos%28y%29%2Bcross%28sin%28x%29sin%28y%29%29%29=+tan%28x%29

%28sin%28x%29cos%28y%29-2sin%28x%29%2Bsin%28x%29cos%28y%29%29%2F%28cos%28x%29cos%28y%29-2cos%28x%29%2Bcos%28x%29cos%28y%29%29=+tan%28x%29

Combine like terms:

%282sin%28x%29cos%28y%29-2sin%28x%29%29%2F%282cos%28x%29cos%28y%29-2cos%28x%29%29=+tan%28x%29

Factor out 2sin%28x%29 in the top and 2cos%28x%29 in the
bottom:

2sin%28x%29%28cos%28y%29-1%29%2F%282cos%28x%29%28cos%28y%29-1%29%29=+tan%28x%29

Cancel the 2's and the %28cos%28y%29-1%29's.

cross%282%29sin%28x%29cross%28%28cos%28y%29-1%29%29%2F%28cross%282%29cos%28x%29cross%28%28cos%28y%29-1%29%29%29=+tan%28x%29

sin%28x%29%2Fcos%28x%29=tan%28x%29

Then use identity sin%28x%29%2Fcos%28x%29=tan%28x%29

tan%28x%29=tan%28x%29

Edwin