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Question

If cosα+cosβ=a,sinα+sinβ=b and αβ=2θ, then cos3θcosθ is equal to:

A
a2+b22
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B
a2+b23
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C
3a2b2
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D
a2+b24
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Solution

The correct option is C a2+b23

cos3θcosθ=4cos2θ3
=2cos2θ1
=2cos(αβ)1 .........(1)
Now a2+b2=(cos2α+cos2β+2cosαcosβ)+(sin2α+sin2β+2sinαsinβ)
=2+2(cosαcosβ+sinαsinβ)
=2+2cos(αβ)
So, 2cos(αβ)=a2+b22 ............(2)

cos3θcosθ=a2+b23 ........From (1) and (2)


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