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= cos²30 - sin²135 + 8 sin45 cos135
= (1/2√3)² - (1/2√2)² + (4 sin(45 + 135) + sin (45-135))
= 3/4 - 2/4 + (4 sin 180 + sin(-90))
= 1/4 + (4 . (0 + (-1)))
= 1/4 + (4 . -1)
= 1/4 - 4
= - 15/4