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Question

If α is a root of equation (2sinxcosx)(1+cosx)=sin2x, β is a root of equation 3cos2x10cosx+3=0 and γ is a root of equation 1sin2x=cosxsinx; 0α,β,γπ2, then sinα+sinβ+sinγ can be equal to

A
143262
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B
56
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C
3+426
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D
1+22
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Solution

The correct option is C 3+426
(2sinxcosx)(1+cosx)=sin2x(2sinxcosx)(1+cosx)=1cos2x(1+cosx)(2sinxcosx1+cosx)=0(1+cosx)(2sinx1)=0
cosx=1 or sinx=12
Since, 0απ2
sinα=12, cosα=32 ...(1)

3cos2x10cosx+3=0(3cosx1)(cosx3)=0cosx=13 (cosx[1,1])cosβ=13, sinβ=223 ...(2)

1sin2x=cosxsinx
sin2x+cos2x2sinxcosx=cosxsinx
(cosxsinx)2=cosxsinx
(cosxsinx)(cosxsinx1)=0
sinx=cosx (or) cosxsinx=1
sinγ=cosγ=12 ...(3)
Or,
cosxsinx=1cosx=1+sinx
cosx=1, sinx=0 (0γπ2)
cosγ=1, sinγ=0 ...(4)

sinα+sinβ+sinγ
=12+223+12 (or) 12+223+0
=14+3262 (or) 3+426

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