Cos x cos y / sin x sin y = cos (x+y) + cos (x-y) / cos (x-y) - cos (x+y) ?

1 Answer
Mar 13, 2018

Please see below.

Explanation:

We know that as cos(x+y)=cosxcosy-sinxsiny and cos(x-y)=cosxcosy+sinxsiny, hence

cos(x+y)/cos(x-y)=(cosxcosy-sinxsiny)/(cosxcosy+sinxsiny)

Now applying componendo-dividendo (for details see note below), we get

(cos(x+y)+cos(x-y))/(cos(x+y)-cos(x-y))=(cosxcosy-sinxsiny+cosxcosy+sinxsiny)/(cosxcosy-sinxsiny-cosxcosy-sinxsiny)

= (2cosxcosy)/(-2sinxsiny)

= -(cosxcosy)/(sinxsiny)

or (cos(x+y)+cos(x-y))/(cos(x-y)-cos(x+y))=(cosxcosy)/(sinxsiny)

What is componendo-dividendo - If a/b=c/d, then adding 1 to each side, we get a/b+1=c/d+1 or (a+b)/b=(c+d)/d. Similarly subtracting 1, we get a/b-1=c/d-1 or (a-b)/b=(c-d)/d. Now dividing (a+b)/b=(c+d)/d by (a-b)/b=(c-d)/d, we get

(a+b)/(a-b)=(c+d)/(c-d). Hence, if a/b=c/d, then (a+b)/(a-b)=(c+d)/(c-d)