If fn-cos-2-cos 2-cos7-2-cos3n-2-2sin-2-sin 2-sin7-2-sin3n-2-2- then which among the following is are CORRECT-

Cosθ2+cos 2θ+cos7θ2+⋯+cos(3n 2)θ2 =cosθ2+cos(θ2+3θ2)+cos(θ2+2(3θ2))+⋯+cos(θ2+(n 1)(3θ2)) =sin n(3θ4)sin(3θ4)×[ cos(θ+(n 1)3θ22)] nbsp; Similarly, sinθ2+sin ...

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Byju's Answer

Standard XIII

Mathematics

Sum of Trigonometric Ratios in Terms of Their Product

If fnθ=cosθ2+...

Question

If $f_{n} (θ) = \frac{cos \frac{θ}{2} + cos 2 θ + cos \frac{7 θ}{2} + \dots + cos \frac{(3 n - 2) θ}{2}}{sin \frac{θ}{2} + sin 2 θ + sin \frac{7 θ}{2} + \dots + sin \frac{(3 n - 2) θ}{2}},$ then which among the following is (are) CORRECT?

f_{1} (\frac{π}{2}) = 1

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f_{3} (\frac{3 π}{16}) = \sqrt{2} + 1

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f_{3} (\frac{3 π}{16}) = \sqrt{2} - 1

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f_{5} (\frac{π}{28}) = \sqrt{2} + 1

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Solution

The correct option is D $f_{5} (\frac{π}{28}) = \sqrt{2} + 1$
$cos \frac{θ}{2} + cos 2 θ + cos \frac{7 θ}{2} + \dots + cos \frac{(3 n - 2) θ}{2} = cos \frac{θ}{2} + cos (\frac{θ}{2} + \frac{3 θ}{2}) + cos (\frac{θ}{2} + 2 (\frac{3 θ}{2})) + \dots + cos (\frac{θ}{2} + (n - 1) (\frac{3 θ}{2}))$
$= \frac{sin n (\frac{3 θ}{4})}{sin (\frac{3 θ}{4})} \times ⎡ ⎢ ⎢ ⎢ ⎣ cos ⎛ ⎜ ⎜ ⎜ ⎝ \frac{θ + (n - 1) \frac{3 θ}{2}}{2} ⎞ ⎟ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎥ ⎦$

Similarly, $sin \frac{θ}{2} + sin 2 θ + sin \frac{7 θ}{2} + \dots + sin \frac{(3 n - 2)}{2} θ = \frac{sin n (\frac{3 θ}{4})}{sin (\frac{3 θ}{4})} \times [sin (\frac{(3 n - 1) θ}{4})]$
$∴ f_{n} (θ) = cot \frac{(3 n - 1) θ}{4}$
$f_{1} (\frac{π}{2}) = cot \frac{π}{4} = 1$
$f_{3} (\frac{3 π}{16}) = cot 2 (\frac{3 π}{16}) = \sqrt{2} - 1$
$f_{5} (\frac{π}{28}) = cot (\frac{14}{4} (\frac{3 π}{16})) = cot \frac{π}{8} = \sqrt{2} + 1$

Suggest Corrections

If fn(θ)=cosθ2+cos2θ+cos7θ2+⋯+cos(3n−2)θ2sinθ2+sin2θ+sin7θ2+⋯+sin(3n−2)θ2, then which among the following is (are) CORRECT?

If $f_{n} (θ) = \frac{cos \frac{θ}{2} + cos 2 θ + cos \frac{7 θ}{2} + \dots + cos \frac{(3 n - 2) θ}{2}}{sin \frac{θ}{2} + sin 2 θ + sin \frac{7 θ}{2} + \dots + sin \frac{(3 n - 2) θ}{2}},$ then which among the following is (are) CORRECT?