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Question

If sin4α+4cos4β+2=42 sinαcosβ;
α,β[0,π], then cos(α+β)cos(αβ) is equal to :

A
2
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B
2
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C
0
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D
1
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Solution

The correct option is B 2
sin4α+4cos4β+2=42 sinαcosβ

Applying A.MG.M. on LHS
sin4α+4cos4β+1+144sin4α×4cos4β×1×1
sin4α+4cos4α+242 sinαcosβ

But given that sin4α+4cos4β+2=42 sinαcosβ
This is possible only if
sin4α=4cos4β=1
sinα=±1 and cosβ=±12
sinα=1 and cosβ=±12 ( α,β[0,π] )
sinβ=12

cos(α+β)cos(αβ)
=2sinαsinβ
=2

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