Cos² 240° + sin² 315° x sec² 30° =

cos 240 = cos (180+60) = - cos 60 = -½
sin 315 = sin(360-45) = -sin 45 = -½√2
sec 30 = 1 / cos 30 = 1 / ½√3 = ⅔√3

cos² 240° + sin² 315° × sec² 30°
= (-½)² + (-½√2)² × (⅔√3)²
= ¼ + ½ × 4/3
= ¼ + ⅔ = (3+8)/12 = 11/12