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Question

Column - 1: Trigonometric expression
Column - 2: Dependency on 'α' and 'β'
Column - 3: Equivalent value of Trigonometric expression in Column - 1.

Column1Column2Column3(I)cos2(α+β)sin2(αβ)cos(2α)cos(2β)+1(i)Independent of α only (P)1(II)cos2α+cos2(α+β)2cos(α)cos(β)cos(α+β)(ii)Independent of β only (Q)sin2β(III)sin(αβ)sin(α+β)1tan2α cot2β(iii)Independent of α and β(R)cos2αsin2β(IV)2sin2β+4cos(α+β)sinαsinβ+cos2(α+β)(S)cos(2α)
Which of the following options is the only correct combination ?

A
(II)(i)(Q)
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B
(II)(iii)(P)
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C
(III)(i)(Q)
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D
(III)(ii)(S)
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Solution

The correct option is A (II)(i)(Q)
(A) (I) 1+cos2(α+β)sin2(αβ)cos(2α)cos(2β)=1+cos2α cos 2βcos2α cos 2β=1

(II) cos2α+cos2(α+β)cos(α+β)[cos(α+β)+cos(αβ)]=cos2α[cos2αsin2β]=sin2β

(III) (sin2αsin2β)cos2α sin2βcos2 αsin2 βsin2αcos2β

=(sin2αsin2β)cos2αsin2βsin(α+β)sin(βα)

=cos2αsin2β

(IV) 2sin2β+2cos(α+β)[cos(αβ)cos(α+β)]+2cos2(α+β)1

=2(sin2β)12 cos2(α+β)+2cos2(α+β)+2(cos2αsin2β)

=2cos2α1=cos2α

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