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Circular Measurement of Angle

cos2 α + β + ...

Question

$c o s^{2} (α + β) + c o s^{2} (α - β) - c o s 2 α c o s 2 β$ =

A

4

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B

3

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C

2

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D

1

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Solution

The correct option is D 1
${cos}^{2} (α + β) + {cos}^{2} (α - β) - cos 2 α cos 2 β$
$\frac{cos 2 (α + β) + 1}{2} + \frac{cos 2 (α - β) + 1}{2} - cos 2 α cos 2 β$
$\frac{cos (2 α + 2 β) + cos (2 α - 2 β) + 2}{2} - cos 2 α cos 2 β$
$= \frac{2 cos (\frac{2 α + 2 β + 2 α - 2 β}{2}) cos (\frac{2 α + 2 β - 2 α + 2 β}{2}) + 2}{1} - cos 2 α cos 2 β$
using $cos C + cos D = 2 cos (\frac{C + D}{2}) cos (\frac{C - D}{2})$
$= cos 2 α cos 2 β + 1 - cos 2 α cos 2 β$
$= 1$

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