tan(pi/4 + a/2) - tan(pi/4 - a/2)

Using the tangent sum formula, we have:

tan(A + B) = (tan A + tan B) / (1 - tanA*tanB)

Therefore, for tan(pi/4 + a/2):

tan(pi/4 + a/2) = (tan(pi/4) + tan(a/2)) / (1 - tan(pi/4) tan(a/2))
= (1 + tan(a/2)) / (1 - 1 tan(a/2))
= (1 + tan(a/2)) / (1 - tan(a/2))

And for tan(pi/4 - a/2):

tan(pi/4 - a/2) = (tan(pi/4) - tan(a/2)) / (1 + tan(pi/4) tan(a/2))
= (1 - tan(a/2)) / (1 + 1 tan(a/2))
= (1 - tan(a/2)) / (1 + tan(a/2))

Substitute these values back into the original expression:

(tan(pi/4 + a/2)) - (tan(pi/4 - a/2))
= (1 + tan(a/2)) / (1 - tan(a/2)) - (1 - tan(a/2)) / (1 + tan(a/2))
= (1 + tan(a/2))^2 - (1 - tan(a/2))^2 / (1 - tan(a/2))(1 + tan(a/2))
= (1 + 2tan(a/2) + tan^2(a/2)) - (1 - 2tan(a/2) + tan^2(a/2)) / (1 - tan^2(a/2))
= 4tan(a/2) / (1 - tan^2(a/2))
= 4tan(a/2) / sec^2(a/2)
= 4sin(a/2) / cos^2(a/2)
= 4sin(a/2) / (1 - sin^2(a/2))
= 4sin(a/2) / cos(a/2)
= 4tan(a/2)

So, the solution is 4tan(a/2).