6.
$\sec ^2 \theta + 2\cos ec^2 \theta = \frac{1}{{\cos ^2 \theta }} + \frac{2}{{\sin ^2 \theta }}$
โดยอสมการ Cauchy
$\left( {\frac{1}{{\cos \theta }}} \right)\left( {\cos \theta } \right) + \left( {\frac{{\sqrt 2 }}{{\sin \theta }}} \right)(\sin \theta ) \le \sqrt {\frac{1}{{\cos ^2 \theta }} + \frac{2}{{\sin ^2 \theta }}} \sqrt {\cos ^2 \theta + \sin ^2 \theta } $
$1 + \sqrt 2 \le \sqrt {\frac{1}{{\cos ^2 \theta }} + \frac{2}{{\sin ^2 \theta }}} $
$3 + 2\sqrt 2 \le \frac{1}{{\cos ^2 \theta }} + \frac{2}{{\sin ^2 \theta }}$
ดังนั้นค่าต่ำสุดคือ $3 + 2\sqrt 2 $
หาได้เมื่อ $\theta = \arctan \left( {2^{\frac{1}{4}} } \right)$