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Trigonometric Ratios of Compound Angles

0 ≤ cos2 α + ...

Question

$0 \leq c o s^{2} α + c o s^{2} (α + β) - 2 c o s α c o s β c o s (α + β) \leq 1$

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Solution

Middle term :
$= c o s^{2} α - c o s (α + β) [c o s α c o s β + s i n α s i n β]$
$= c o s^{2} α - c o s (α + β) c o s (α - β)$
$= c o s^{2} α - (c o s^{2} α - s i n^{2} β) = s i n^{2} β$
which lies between 0 and 1

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