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Question Number 187311 by anurup last updated on 16/Feb/23
If f_k (x)=(1/k)(sin^k x+cos^k x) find f_4 (x)−f_6 (x)  f_4 (x)−f_6 (x)=(1/4)(sin^4 x +cos^4 x)−(1/6)(sin^6 x+cos^6 x)     =(1/2)sin^4 x((1/2)−(1/3)sin^2 x)+(1/2)cos^4 x((1/2)−(1/3)cos^2 x)
$$\mathrm{If}\:{f}_{{k}} \left({x}\right)=\frac{\mathrm{1}}{{k}}\left(\mathrm{sin}\:^{{k}} {x}+\mathrm{cos}\:^{{k}} {x}\right)\:\mathrm{find}\:{f}_{\mathrm{4}} \left({x}\right)−{f}_{\mathrm{6}} \left({x}\right) \\ $$$${f}_{\mathrm{4}} \left({x}\right)−{f}_{\mathrm{6}} \left({x}\right)=\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{sin}\:^{\mathrm{4}} {x}\:+\mathrm{cos}\:^{\mathrm{4}} {x}\right)−\frac{\mathrm{1}}{\mathrm{6}}\left(\mathrm{sin}\:^{\mathrm{6}} {x}+\mathrm{cos}\:^{\mathrm{6}} {x}\right) \\ $$$$\:\:\:=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}\:^{\mathrm{4}} {x}\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{3}}\mathrm{sin}\:^{\mathrm{2}} {x}\right)+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:^{\mathrm{4}} {x}\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{3}}\mathrm{cos}\:^{\mathrm{2}} {x}\right) \\ $$
Commented by anurup last updated on 16/Feb/23
Can anyone help me to solve this
$$\mathrm{Can}\:\mathrm{anyone}\:\mathrm{help}\:\mathrm{me}\:\mathrm{to}\:\mathrm{solve}\:\mathrm{thi}{s} \\ $$
Commented by anurup last updated on 16/Feb/23
Can anybody tell me how to proceed further?
$$\mathrm{Can}\:\mathrm{anybody}\:\mathrm{tell}\:\mathrm{me}\:\mathrm{how}\:\mathrm{to}\:\mathrm{proceed}\:\mathrm{further}? \\ $$

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