If u_n=sin^n(theta)+cos^n(theta) and (u_3-u_5)/(u_1)=K((u_5-u_7)/(u_3)) then find K?

1 Answer
Jul 21, 2018

K=1.

Explanation:

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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