For two angles a and b tan(a - b) = (tan a - tan b)/(1 + tan a*tan b)

Now in (tan(x+y) - tan y)/(1 + tan(x+y)*tan y) take x + y as a and y as b

=> (tan a - tan b)/(1 + tan a*tan b)

This is...

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For two angles a and b tan(a - b) = (tan a - tan b)/(1 + tan a*tan b)

Now in (tan(x+y) - tan y)/(1 + tan(x+y)*tan y) take x + y as a and y as b

=> (tan a - tan b)/(1 + tan a*tan b)

This is the same as tan (a - b)

= tan(x + y - y)

= tan x

**This proves that (tan(x+y) - tan y)/(1 + tan(x+y)*tan y) = tan x**