Why are the tangents for 90 and 270 degrees undefined?

I'll try to give you a visual explanation.
Consider the trigonometric meaning of tangent of an angle #alpha#:

The tangent of #alpha# is equal to the length of the segment #AB#; but when #alpha# becomes #90^@# the length of #AB# get stretched upwards (or downwards for #alpha=270^@#) so that we will never meet #A#!!!

Hope it helps!

You can also say that #tanx = sinx/cosx#.

You can take a look at the overlap of #sinx# and #cosx# below to see what happens to each as we approach #90^o# and #270^o# from the right or left:

graph{(y - sinx)(y - cosx) = 0 [-0.034, 6.2831, -1.2, 1.2]}

We can establish that:

#lim_(x->90^(o^(-))) sinx/cosx = -1/0 = -oo#

because #cosx# decreases while #sinx# increases as #x->90^o# from the left, and

#lim_(x->90^(o^(+))) sinx/cosx = 1/0 = oo#

because both #cosx# and #sinx# increase as #x->90^o# from the right.

We can also see that:

#lim_(x->270^(o^(-))) sinx/cosx = -1/0 = -oo#

because #sinx# decreases but #cosx# increases as #x->270^o# from the left, and

#lim_(x->270^(o^(+))) sinx/cosx = 1/0 = oo#

because both #sinx# and #cosx# decrease as #x->270^o# from the right.

Since the limits from the left and right side are not the same, #\mathbf(tanx)# of those angles is undefined.