 Solution

Expand $(\cos \theta +i\sin \theta {)}^4$ using De Moivre's theorem, then divide its imaginary and real parts by ${\cos }^4\theta$. With $u=\tan \theta$, this givesThe quotient holds where its denominator and $\cos \theta$ are nonzero. At $\theta =\pi /8$, use the undivided real-part identity: $\cos 4\theta =0$ implies $u^4-6u^2+1=0$. Thus $u^2=3\pm 2\sqrt{2}$. Since $0<\tan (\pi /8)<1$, the smaller positive value applies:This limiting step avoids treating an undefined value of $\tan 4\theta$ as an ordinary finite number.