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If un=sin^("n")theta+cos^ntheta, then pr...

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

Text Solution

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`(u_5-u_(7))/(u_3-u_(5))=((sin^5theta +cos^5theta)-(sin^7theta+cos^7theta))/((sin^3theta +cos^3theta)-(sin^5theta+cos^5theta))`
`(sin^5theta(1-sin^2theta)+cos^5theta(1-cos^2theta))/(sin^3theta(1-sin^2theta)+cos^3theta(1-cos^2theta))`
`=(sin^2thetacos^2theta[sin^3theta+cos^3theta])/(sin^2thetacos^2theta[sintheta+costheta])=u_3/u_1`
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