Given that, u_n=sin^ntheta+cos^ntheta.
:. u_3-u_5=(sin^3theta+cos^3theta)-(sin^5theta+cos^5theta),
=(sin^3theta-sin^5theta)+(cos^3theta-cos^5theta),
=sin^3theta(1-sin^2theta)+cos^3theta(1-cos^2theta),
=sin^3thetacos^2theta+cos^3thetasin^2theta,
=sin^2thetacos^2theta(sintheta+costheta),
=u_1(sin^2thetacos^2theta).
rArr (u_3-u_5)/u_1=sin^2thetacos^2theta..........(ast^1).
Again, u_5-u_7=sin^5theta+cos^5theta-sin^7theta-cos^7theta,
=sin^5theta(1-sin^2theta)+cos^5theta(1-cos^2theta),
=sin^5thetacos^2theta+cos^5thetasin^2theta,
=sin^2thetacos^2theta(sin^3theta+cos^3theta),
=u_3(sin^2thetacos^2theta).
rArr (u_5-u_7)/u_3=sin^2thetacos^2theta.........(ast^2).
"Hence, "(u_3-u_5)/u_1=K((u_5-u_7)/u_3), (ast^1) and (ast^2),
rArr K=1.
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