Question

If x = cos 1° cos 2° cos 3° …. cos 89° and y = cos 2° cos 6° cos 10° … cos 86° and , where a and b are co-prime, then find . A+b/5

Answer 1

Answer:

If x=cos1cos2cos3..cos89 and y=cos2cos6cos10…cos86, then what is the nearest integer to 2/7log (base2) (y/x)?

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Given,

x=cos(1)cos(2)cos(3)…cos(87)cos(88)cos(89)

and y=cos(2)cos(6)cos(10)…cos(82)cos(86)

As we know,

cos(A)cos(B)=cos(A+B)+cos(A−B)2

So,

cos(1)cos(89)=cos(89+1)+cos(89−1)2=cos(90)+cos(88)2

As, cos(90)=0

cos(1)cos(89)=cos(88)2

Similarly,

cos(2)cos(88)=cos(88+2)+cos(88−2)2=cos(86)2

and so on.

So,

x=cos(1)cos(2)cos(3)…cos(87)cos(88)cos(89)

=cos(88)cos(86)cos(84)…cos(4)cos(2)cos(45)244

=cos(88)cos(86)cos(84)…cos(4)cos(2)2442–√

[As cos(45)=12√ ]

Now,

cos(2)cos(88)=cos(86)2 [Similar analogy as above.]

So,

x=cos(88)cos(86)cos(84)…cos(4)cos(2)2442–√

=cos(86)cos(82)cos(78)…cos(6)cos(2)2662–√

=y2662–√

=y266+12

=y21332

So,

2log2(yx)7

=2log2(21332)7

=2×133)7×2

=19