1.

Let sec ^(2)((pi)/(9))+sec^(2)((2pi)/(9))+sec_(2)((4pi)/(9))=S and Sigma_(k-1)^(89) cos^(6)(k^(o))=(a)/(b) ,where a,b are coprime, then which of the following is/are correct?

Answer»

Number of positive divisors of a+B + S is 4
Number of positive divisors of a+b + S is 8
S is a perfect square
a+b is a prime number

Solution :`sec^(2)((PI)/(9))+sec^(2)((2pi)/(9))+sec^(2)((4pi)/(9))`
`=((cos.(2pi)/(9))^(2)(cos.(4pi)/(9))^(2) +(cos.(4pi)/(9))^(2)(cos.(pi)/(9))^(2)+cos^(2)((pi)/(9))cos^(2)((2pi)/(9)))/(cos.(pi)/(9)cos.(2pi)/(9)cos.(4pi)/(9))^(2)`
Using CD formulas
`=(1)/(4)([(cos.(2pi)/(3)+cos.(2pi)/(9))^(2)+(cos.(5pi)/(9)+cos.(pi)/(3))^(2)+(cos.(pi)/(3)+cos.(pi)/(9))^(2)])/(((1)/(64)))`
`= 16 ((3)/(4)+cos^(2).(pi)/(9)+cos^(2).(2pi)/(9)+cos^(2).(4pi)/(9)+cos.(pi)/(9)-cos^(2).(2pi)/(9)-cos.(4pi)/(9))`
`=16((3)/(9)+(3)/(2))`
`therefore cos .(pi)/(9)-cos^(2).(2pi)/(9)-cos.(4pi)/(9)=0 =36`
`underset(k=1)overset(89)Sigmacos^(6)(k^(@))= underset(k=1)overset(89)Sigmasin^(6)(k^(@))=(1)/(2)underset(k=1)overset(89)sum(SIN^(6)(k^(@))+cos^(6)(k^(@)))`
`(1)/(2)(underset(k=1)overset(89)Sigma(1-(3)/(4)sin^(2)(2k^(@))))=(89)/(2)-(3)/(8)underset(k=1)overset(89)Sigma(2k)^(@)`
`=(89)/(2)-(3)/(8)xx45`
`(a)/(b)=(221)/(8)`
`a221,b=8`


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