Let [f_k( x ) = 1 k( ^kx + ^kx )] where [x R] and [k ,1.] Then [f_4( x ) - f_6( x ),,] equ…

Question

Let $$f_k\left( x \right) = {1 \over k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right)$$ where $$x \in R$$ and $$k \ge \,1.$$

Then $${f_4}\left( x \right) - {f_6}\left( x \right)\,\,$$ equals :

Accepted Answer

Let $${f_k}\left( x \right) = {1 \over k}\left( {{{\sin }^k}x + {{\cos }^k}.x} \right)$$

Consider

$${f_4}\left( x \right) - {f_6}\left( x \right)  $$

$=$ $${1 \over 4}\left( {{{\sin }^4}x + {{\cos }^4}x} \right) - {1 \over 6}\left( {{{\sin }^6}x + {{\cos }^6}x} \right)$$

$$ = {1 \over 4}\left[ {1 - 2{{\sin }^2}x{{\cos }^2}x} \right] - {1 \over 6}\left[ {1 - 3{{\sin }^2}x{{\cos }^2}x} \right]$$

$$ = {1 \over 4} - {1 \over 6} = {1 \over {12}}$$

Solutions · 1