Left 1-cos frac 4 k -1 pi 4 k right - then which of the following is-are correct- A- P 3-3-16B- P 4-2-2-16- P 5-3-5-32D- P 6-2-3-16

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Special Integrals - 2

left 1+cos fr...

Question

Let $P (k) = (1 + c o s \frac{π}{4 k}) (1 + c o s \frac{(2 k + 1) π}{4 k}) (1 + c o s \frac{(2 k - 1) π}{4 k}) (1 + c o s \frac{(4 k - 1) π}{4 k})$ then which of the following is/are correct?

P (3) = \frac{3}{16}

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P (4) = \frac{2 - \sqrt{2}}{16}

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P (5) = \frac{3 - \sqrt{5}}{32}

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P (6) = \frac{2 - \sqrt{3}}{16}

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Let P(x) = x⁶ + ax⁵ + bx⁴ + cx³ + dx² + ex + f be a polynomial such that P(1)=1; P(2)=2; P(3)=3; P(4)=4; P(5)=5; P(6)=6 then find the value of P(7).

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}},

P (1) = 1

P (2) = 2

P (3) = 3

P (4) = 5

P (6)

{(1 + x)}^{n} = P_{0} + P_{1} x + P_{2} x^{2} + . . . + P_{n} x^{n}

\frac{p_{0} - p_{2} + p_{4} - p_{6} + . . .}{p_{1} - p_{3} + p_{5} - p_{7} + . . .} = ?

{cos}^{2} (\frac{π}{16}) + {cos}^{2} (\frac{3 π}{16}) + {cos}^{2} (\frac{5 π}{16}) + {cos}^{2} (\frac{7 π}{16})

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