1.

If (n,theta)=prod_(n=1)^(n)((1+tan^(2)(2""^(n)theta))/((1-tan^(2)(2""^(n)theta))^(2))) then find the value of 8f (3,(pi)/(8))?

Answer»

SOLUTION :`f(n,THETA)=underset(n=1)overset(n)prod(((1+tan^(2)(2^(n)theta))/(1tan^(2)(2^(n)theta)))^(2).(1)/(a+tan^(2)(2^(n)theta)))=underset(n=1)overset(n)prod(COS^(2)(2^(n)theta))/(cos^(2)(2.2^(n)theta))`
`=(cos^(2)2theta)/(cos^(2)2^(2)theta).(cos^(2)2^(2)theta)/(cos^(2)2^(3)theta).(cos^(2).2^(3)theta)/(cos2^(2).2^(4)theta).........(cos^(2)2^(n)theta)/(cos^(2)2^(n+1)theta)`
`f(n,theta)=(cos^(2)2 theta)/(cos^(2)2^(n+1)theta)`
`f(3,(PI)/(8))=(cos^(2)""(pi)/4)/(cos^(2)(2^(4).(pi)/(8)))=((1)/(sqrt2))^(2)/((1)^(2))=1/2`
So, `8f(3,(pi)/(8))=4.Ans."]"`


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