If `acos^2theta+3acosthetas in^2theta=ma n das in^3theta+3acos^2thetas in^theta=n` , then prove that: `(m+n)^(2//3)+(m-n)^(2//3)=2a^(2//3)`

Explanation

Given : `a cos ^{3} theta+3 a cos theta sin ^{2} theta=m ldots . .(1)`

`a sin ^{3} theta+3 a cos ^{2} theta sin theta=n . ldots . .(2)`

Adding equation (1) and (2) we get,

`therefore m+n=a(cos ^{3} theta+sin ^{3} theta)+3 a cos theta sin theta(cos theta+sin theta)=a(cos theta+sin theta)^{3}`

Similarly, we get

`m-n=a(cos theta-sin theta)^{3}`

`=(m+n)^{frac{2}{3}}+(m-n)^{frac{2}{3}}`

`=a^{2 / 3}[((cos theta+sin theta)^{3})^{2 / 3}+((cos theta-sin theta)^{3})^{2 / 3}]`

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