# Exclusions found but Solve and Reduce don't find exclusions

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

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


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

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

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


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

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

• 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] Jul 26 '15 at 16:55
• @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? Jul 26 '15 at 17:07

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