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.