Show that : `sin^(2n) x+cos^(2n) xle1.` n belongs to N

If tan x + cot x =2 , then sin^(2n)x + cos^(2n) x is equal to

If 0 lt x lt pi/2 and sin^(n) x + cos^(n) x ge 1 , then

Prove by the principle of mathematical induction that for all n belongs to N :\ 1/(1. 3)+1/(3.5)+1/(5.7)++1/((2n-1)(2n+1))=n/(2n+1)

Prove that sin (n+1) x sin (n +2) x + cos (n +1) x cos (n +2) x = cos x

Prove the following by the principle of mathematical induction: 1/2tan(x/2)+1/4tan(x/4)+...+(1/2^n)tan(x/(2^n))=(1/2^n)cot(x/(2^n)) - cot x for all n belongs to N.

Evaluate : intx^(2n-1)sin(x^(n))dx

Prove that: sin (n + 1) x sin (n + 2)x + cos (n + 1) x cos (n + 2) x = cos x

Prove that cos^2 x + cos^2 3x + cos^2 5x + ....+ to n terms= 1/2 [ n+ (sin4nx)/(2 sin2x)]

Let f(x)=lim_(n to oo) ((2 sin x)^(2n))/(3^(n)-(2 cos x)^(2n)), n in Z . Then

If sinx+cos e cx=2, then sin^n x+cos e c^n x is equal to 2 (b) 2^n (c) 2^(n-1) (d) 2^(n-2)