Question

If $c{\cos }^3\theta +3c\cos \theta {\sin }^2\theta =m$,
$c{\sin }^3\theta +3c{\cos }^3\theta \sin \theta =n$, then prove that
${(m+n)}^{2/3}+{(m-n)}^{2/3}=2c^{2/3}$.

Answer 1

Adding the given relation, we get
$c{(\sin \theta +\cos \theta )}^3=m+n$
or $c^{2/3}{(\sin \theta +\cos \theta )}^3={(m+n)}^{2/3}$.
Similarly, by subtracting, we shall get
$c^{2/3}{(\sin \theta +\cos \theta )}^2={(m+n)}^{2/3}$
Now add $\left(1\right)$ and $\left(2\right)$.

${(m+n)}^{2/3}+{(m-n)}^{2/3}=2c^{2/3}$.