Let `P(k)=(1+cos(pi/(4k)))` `(1+cos(((2k-1)pi)/(4k)))` `(1+cos(((2k+1)pi)/(4k)))(1+cos(((4k-1)pi)/(4k)))dot` Then Prove that (a)`P(3)=1/(16)` (b) `P(4)=(2-sqrt(2))/(16)` (c) `P(5)=(3-sqrt(5))/(32)` (d) `P(6)(2-sqrt(3))/(16)`

Explanation

`P(k) = (1+cos(pi/(4k)))(1+cos(pi/2-pi/(4k)))(1+cos(pi/2+pi/(4k)))(1+cos(pi-pi/(4k)))`

`=>P(k) = (1+cos(pi/(4k)))(1+sin(pi/(4k)))(1-sin(pi/(4k)))(1-cos(pi/(4k)))`

`=>P(k) = (1-cos^2(pi/(4k))) (1-sin^2(pi/(4k)))`

`=>P(k) = sin^2(pi/(4k))cos^2(pi/(4k))`

`=>P(k) = 1/4sin^2(pi/(2k))`

Now, `P(3) = 1/4sin^2(pi/6) = 1/4*1/4 = 1/16`

`P(4) = 1/4sin^2(pi/8) = 1/4((1-cos(pi/4))/2) = 1/8(1-1/sqrt2) = (2-sqrt2)/16`

`P(5) = 1/4sin^2(pi/10) = 1/4((sqrt5-1)/4)^2 = (3-sqrt5)/32`

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