Question

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

Answer 1

$m+n=c\left[{\sin }^3\theta +{\cos }^3\theta +3\sin \theta \cos \theta \left(\sin \theta +\cos \theta \right)\right]$

$\Rightarrow m+n=c{\left(\sin \theta +\cos \theta \right)}^3$

$\Rightarrow {\left(m+n\right)}^{\frac{a}{3}}=c^{\frac{a}{3}}{\left(\sin \theta +\cos \theta \right)}^a$

and $m-n=c\left[{\cos }^3\theta -{\sin }^3\theta -3\cos \theta \sin \theta \left(\cos \theta -\sin \theta \right)\right]$

$\Rightarrow m-n=c{\left(\cos \theta -\sin \theta \right)}^3$

$\Rightarrow {\left(m-n\right)}^{\frac{a}{3}}=c^{\frac{a}{3}}{\left(\cos \theta -\sin \theta \right)}^a$

Now, $(m+n{)}^{\frac{a}{3}}+(m-n{)}^{\frac{a}{3}}=c^{\frac{a}{3}}\left[{\left(\sin \theta +\cos \theta \right)}^a+{\left(\cos \theta -\sin \theta \right)}^a\right]$

If $a=2$

Then, $(m+n{)}^{\frac{a}{3}}+(m-n{)}^{\frac{a}{3}}=2c^{\frac{a}{3}}$

Therefore $a=2$