Question 756774
I am confused on the tan(x-y) if tan x = 24/7 and tan y = -4/3...how is tan (x-y) = -4/3?
<pre>
I agree it seems strange that tan(x-y) and tan(y) are the same, but it's
a fact here.  [It has to do with the fact that both angles are from integer right triangles.  x is from a 7-24-25 right triangle and y is from a 3-4-5
right triangle.

Use the identity:

tan(x-y) = {{{(tan(x)-tan(y))/(1+tan(x)tan(y))}}}

tan(x-y) = {{{(tan(x)-tan(y))/(1+tan(x)tan(y))}}} = {{{(24/7-(-4/3))/(1+(24/7)(-4/3))}}} = {{{(24/7+4/3)/(1-(24/7)(4/3))}}} = {{{(24/7+4/3)/(1-(""^8cross(24)/7)(4/cross(3)))}}} = {{{((24*3)/(7*3)+(4*7)/(3*7))/(1-(8/7)(4))}}} =  {{{(72/21+28/21)/(1-32/7)}}} = {{{(100/21)/(7/7-32/7)}}} = 

{{{(100/21)/(-25/7)}}} = {{{(100/21)*(-7/25)}}} = {{{-(100/21)*(7/25)}}} = {{{-(""^4cross(100)/21)*(7/cross(25))}}} = {{{-(""^4cross(100)/(""^3cross(21)))*(cross(7)/cross(25))}}} = {{{-4/3}}}

Since it seems strange that both tan(y) and tan(x-y) can both be {{{-4/3}}}.  let's check it with a calculator:

x = 73.73979529°, y=126.8398976°

x-y = 73.73979529° - 126.8398976° = -53.13010235°

tan(y) = tan(126.8398976°) = -1.333333333

tan(x-y) = tan(-53.13010235) = -1.333333333

So you see that, strange as it seems, this is the case!

Edwin</pre>