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I have the following from Stan Wagons Mathematica in Action.

Plot[Sec[x]/(1 + x^2 Tan[x]), {x, -((5 \[Pi])/2), (5 \[Pi])/2},
 Exclusions -> {Cos[x] == 0, 1 + x^2 Tan[x] == 0},
 ExclusionsStyle -> {{Dashing[0.02], Gray, Thickness[0.008]}},
 FrameTicks -> {{Automatic, None}, {Range[-2 Pi, 2 Pi, Pi], None}}]

Which produced:

enter image description here

However, neither Solve nor Reduce seem to find the Exclusions:

In[79]:= Reduce[1 + x^2 Tan[x] == 0, x, Reals]

During evaluation of In[79]:= Reduce::nsmet: This system cannot be solved with the methods available to Reduce. >>

Out[79]= Reduce[1 + x^2 Tan[x] == 0, x, Reals]

In[77]:= Solve[1 + x^2 Tan[x] == 0, x, Reals]

During evaluation of In[77]:= Solve::nsmet: This system cannot be solved with the methods available to Solve. >>

Out[77]= Solve[1 + x^2 Tan[x] == 0, x, Reals]

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    $\begingroup$ As there are infinitely many zeroes, you have to limit it to the same range in your Plot[]: Reduce[1 + x^2 Tan[x] == 0 && -5 π/2 < x < 5 π/2, x, Reals] $\endgroup$ – J. M. is away Jul 26 '15 at 16:55
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    $\begingroup$ @Guesswhoitis. Thanks, that works. I had just thought of trying that before your comment, but thanks for the help. Any way to make the Cos[x]==0 exclusions a different ExclusionsStyle? $\endgroup$ – David Jul 26 '15 at 17:07
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The equation has infinite number of nontrivial roots. You can query a subset of them, though:

Solve[1 + x^2 Tan[x] == 0 && -10 < x < 10, x, Reals]

(* {{x -> Root[{1 + #1^2 Tan[#1] &, -9.4360086156910331017}]},
    {x -> Root[{1 + #1^2 Tan[#1] &, -6.3083089552381513776}]},
    {x -> Root[{1 + #1^2 Tan[#1] &, -3.2367552992046412986}]},
    {x -> Root[{1 + #1^2 Tan[#1] &, -0.89520604538423185008}]},
    {x -> Root[{1 + #1^2 Tan[#1] &, 3.0333351651192716892}]},
    {x -> Root[{1 + #1^2 Tan[#1] &, 6.2576534384926035492}]},
    {x -> Root[{1 + #1^2 Tan[#1] &, 9.4134935234375590912}]}} *)
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