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Question

Match the column

Column AColumn B1.cos2π7 + cos4π7 + cos6π7 A.0 2.cosπ7 + cos2π7 +cos3π7 + cos4π7 + cos5π7 + cos6π7 B.123.sinπ11 + sin3π11 + sin5π11 + sin7π11 + sin9π11 C.12


A

1 - A, 2 - B, 3 - C

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B

1 - C, 2 - A, 3 - B

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C

1 - A, 2 - C, 3 - B

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D

1 - B, 2 - A, 3 - C

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Solution

The correct option is B

1 - C, 2 - A, 3 - B


We know, cosα + cos(α+β) + cos(α+2β) + .......... cos(α+(n1)β)

just apply that formula

α = 2π7

β = 2π7

n = 3

= sin(3×2π2×7)sin(2π2×7)×cos(2π7+(31)2×2π7)

= sin(3π7)×cos(4π7)sin(π7)

= sin3π7.cos(π3π7)sin(π7) [cos (πθ) = - cos θ]

= sin3π7[cos3π7]sinπ7

= sin3π7.cos3π7sinπ7

We need to further simplify the expression, we observe that numerator is in the form of sin θ × cosθ.Apply the formula sin2θ = 2sinθ × cosθ

Multiplying 2 in numerator and denominator

We observe that we can have same trigonometric function in numerator and denominator by writing numerator as sin (πθ) form because sin (πθ) = sin θ

we Know,

sin6π14×0sinπ14 = 0

3. cosπ11 + cos3π11 + cos5π11 + cos7π11 + cos9π11

= π11, β = 2π11,n=5

= sin(5×2π2×11)sin(2π2×11).cos(π11+4×2π2×11)

Multiply and divide by 2 in the above expression.


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