How to prove a transcendental equation is never zero or sometimes zero?

Solution 1:

Since:

  • $f$ is continuous;
  • $f\left(\frac\pi2\right)=1+\frac14\pi^2\sinh\left(\frac\pi2\right)>0$;
  • $f\left(\frac{3\pi}2\right)=1-\frac94\pi^2\sinh\left(\frac{3\pi}2\right)<0$,

$f$ has a zero in the interval $\left(\frac\pi2,\frac{3\pi}2\right)$, by the intermediate value theorem.