Question

tan(45+a)-tan(45-a)

Answer 1

\[={(1+tana)/1-tana}-{(1-tana)/1+tana}\]\[{(1+tana)^2-(1-tana)^2}/1-\tan^2a\]\[=4tana/1-\tan^2a\]

Answer 2

Scroll down to "The Other Identities" section at the following web site:
http://library.thinkquest.org/20991/alg2/trigi.html

There you will find these identities for Tan(A+B) and Tan(A-B):
tan (A + B) = (tan A + tan B)/(1 - (tan A)(tan B))
tan (A - B) = (tan A - tan B)/(1 + (tan A)(tan B))

From the problem it is assumed that tan(45) is the tangent of 45 degrees, which is 1.

Thus

tan(45+a)-tan(45-a) ->
\[(1+\tan a)/(1-\tan a)-(1-\tan a/(1+\tan a)\]
When he above expression is combined and simplified, the result is:
\[2 \tan 2a\]

Note: If any one can show that final result is invalid, I'm certain that Wolfram Research would be glad to hear from you.