Calculate tan a and tan b confusion

[Loki123]

If tan a + tan b = 2
Tan (a+b) = 4
And we know tan a is less than tan b, determine tan a and tan b.
Under "apparently" I have written solutions. My problem is that I do not understand how I can just use what I got for tan b. I get that the solution with + goes to tan b because it's bigger than tan a, but when I plug the answers back into the equation for tan a I get something different. It just doesn't make sense to me. How??

 

[Deleted member 4993]

If tan a + tan b = 2
Tan (a+b) = 4
And we know tan a is less than tan b, determine tan a and tan b.
Under "apparently" I have written solutions. My problem is that I do not understand how I can just use what I got for tan b. I get that the solution with + goes to tan b because it's bigger than tan a, but when I plug the answers back into the equation for tan a I get something different. It just doesn't make sense to me. How??
View attachment 31643

If you use tan(ß) = (2 + √2)/2 →

tan(α) = 1/(2*tan(ß)) = 1/(2+√2) =1/(2+√2) * (2-√2)/(2-√2) = (2-√2)/(2^2-2) = (2-√2)/2

 

[Loki123]

If you use tan(ß) = (2 + √2)/2 →

tan(α) = 1/(2*tan(ß)) = 1/(2+√2) =1/(2+√2) * (2-√2)/(2-√2) = (2-√2)/(2^2-2) = (2-√2)/2

thank you ♥