\begin{align}
& \frac{ (1-\cos^{2}x)^2 }{a}+\frac{ (\cos^{2}x)^2 }{b}= \frac{1}{a+b}\\
& \Longrightarrow \frac{ 1-2\cos^{2}x+\cos^{4}x }{a}+\frac{ (\cos^{2}x)^2 }{b}= \frac{1}{a+b}\\
& \Longrightarrow \color{blue}{\frac{a+b}{ab}}(\cos^{2}x)^2\color{blue}{- \frac{2}{a}}\cos^{2}x\color{blue}{+ \frac{b}{a(a+b)}}=0
\end{align}
چون دلتای این معادله صفر میشود پس:
$$\cos^{2}x= \frac{ \frac{2}{a} }{ \frac{2(a+b)}{ab} }=\frac{b}{a+b} \Rightarrow \cos x= \mp \frac{ \sqrt{b} }{\sqrt{a+b}} \Rightarrow \sin x= \mp \frac{ \sqrt{a} }{\sqrt{a+b}} $$
$$ \frac{\sin x}{a^3}+\frac{\cos x}{b^3}=\frac{\mp \frac{ \sqrt{a} }{\sqrt{a+b}}}{a^3}+\frac{\mp \frac{ \sqrt{b} }{\sqrt{a+b}}}{b^3}= \color{red}{\mp \frac{ a^{\frac{-5}{2}}+b^{\frac{-5}{2}}}{\sqrt{a+b}}} $$
