<s>[INST] Find the Value of Trigonometric Expression Given $$\begin{align} \frac{\cos \alpha}{\cos \beta}+\frac{\sin \alpha}{\sin \beta}=-1 \end{align} \tag{1}$$ Find the value of
$$\begin{align} \frac{\cos^3 \beta}{\cos \alpha}+\frac{\sin ^3\beta}{\sin \alpha} \end{align} \tag{2} $$
I Tried like this: From $(1)$ we have$$\sin\alpha \cos\beta+\cos\alpha \sin\beta=-\sin\beta \cos\beta$$ $\implies$
$$\sin(\alpha+\beta)=-\sin\beta \cos\beta \tag{3}$$
Eq $(2)$ is $$\frac{\frac{\sin\alpha}{4}\left(3\cos\beta+\cos3\beta\right)+\frac{\cos\alpha}{4}\left(3\sin\beta-\sin3\beta\right)}{\sin\alpha \cos\alpha}=\frac{\frac{3}{4}\sin(\alpha+\beta)+\frac{1}{4}\sin(\alpha-3\beta)}{\sin\alpha \cos\alpha}=\frac{\frac{-3}{4}\sin\beta \cos\beta+\frac{1}{4}\sin(\alpha-3\beta)}{\sin\alpha \cos\alpha}$$
I cannot proceed any further..please help me 
 [/INST] $$\sin(\alpha+\beta)=-\frac{\sin2\beta}2$$
$$\frac{\sin(\alpha+\beta)}{\sin2\beta}=\frac{-1}2$$
Apply Componendo and dividendo and Prosthaphaeresis Formula  to $$\frac34\sin(\alpha+\beta)+\frac14\sin(\alpha-3\beta)=0$$
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