If `u_n=sin^("n")theta+cos^ntheta,` then prove that `(u_5-u_7)/(u_3-u_5)=(u_3)/(u_1)` .

If u_n=sin^("n")theta+cos^ntheta, then prove that (u_4-u_6)/(u_2-u_4)=1/2 .

If u_(n)=sin^(n)alpha+cos^(n)alpha , then prove that 2u_(6)-3u_(4)+1=0 .

If u_(n)=2 cos n theta , then show that u_(n+1)= u_(1)u_(n)-u_(n-1) , hence, show that 2 cos 5 theta= u_(1)^(5)- 5u_(1)^(3)+ 5u_(1)

If U_n=2 cosntheta then U_1U_n-U_(n-1) is equal to

If U_n=(sqrt(3)+1)^(2n)+(sqrt(3)-1)^(2n) , then prove that U_(n+1)=8U_n-4U_(n-1)dot

A train of mass M is in motion. When its velocity is u_1 and acceleration is a_1 , resistive force against its speed is R_1 . Also when velocity is u_2 and acceleration is a_2 , resistive forced is R_2 . If the engine works at a fixed rate P, show that P(u_2-u_1)=u_1u_2(R_1-R_2)+Mu_1u_2(a_1-a_2) .

If u_(n)=int_(0)^((pi)/(4))tan^(n)xdx , show that, u_(n)+u_(n-2)=(1)/(n-1)(ngt1) , hence find u_(5)

If u_n=sin ^n alpha+cos^nalpha then show that, the value of 6u_4-4u_6 is independent of alpha .

If n is a positive integer and U_(n) = (3 + sqrt5)^(n) + (3 - sqrt5)^(n) , then prove that U_(n + 1) = 6U_(n) - 4U_(n -1), n ge 2