f(n)=sin^(4)n+cos^(4)n

Let n be a natural number such that n gt 4 and U_(n) =sin ^(n) x+ cos ^(n)x. Then which of the following is correct ?

If P_(n)=cos^(n)theta+sin^(n)theta and Q_(n)=cos^(n)theta-sin^(n)theta then show that p_(n-2)=-sin^(2)theta cos^(2) theta p_(n-4) hence show that p_(4)=1-2 sin^(2) theta cos^(2) theta Q_(4)=cos^(2) theta- sin^(2) theta

If f_(n)(x) = (sin x)/(cos3x)+(sin 3x)/(cos 3^(2)x) +(sin 3^(2)x)/(cos 3^(3)x) +....+ (sin 3^(n-1)x)/(cos 3^(n)x)"Then" f_(2) ((pi)/(4)) + f_(3) ((pi)/(4))=

If (sin^4 theta)/a+ (cos^4 theta)/b= 1/(a+b) , prove that (sin^(4n) theta)/a^(2n-1)+ (cos^(4n) theta)/b^(2n-1)= 1/(a+b)^(2n-1), n in N .

The value of lim_(n rarr oo)(1)/(n^(2)){(sin^(3)pi)/(4n)+2(sin^(3)(2 pi))/(4n)+...+n(sin^(3)(n pi))/(4n)} is equal to

Let f:N rarr R and g:N rarr R be two functions and f(1)=08,g(1)=0.6f(n+1)=f(n)cos(g(n))-g(n)sin(g(n)) and g(n+1)=f(n)sin(g(n))+g(n)cos(g(n)) for n>=1.lim_(n rarr oo)f(n) is equal to

Let f: N -> R and g : N -> R be two functions and f(1)=0.8, g(1)=0.6 , f(n+1)=f(n)cos(g(n))-g(n)sin(g(n)) and g (n+1)=f(n) sin(g(n))+g(n) cos(g(n)) for n>=1 . lim_(n->oo) f(n) is equal to