<s>[INST] Trigonometric problem : Eliminate $\theta$ and $\phi$ from the relation and find relation between p and q Question : 
Eliminate $\theta$ and $\phi$ from the relation 
$$\begin{align}
p \cot^2\theta + q \cot^2\phi &= 1 &(1)\\
p \cos^2\theta + q \cos^2\phi &= 1 &(2)\\ 
p \sin\theta  &= q\sin\phi &(3)
\end{align}$$
Also find relation between $p$ and $q$.
I have tried different ways but unable to eliminate $\theta$ and $\phi$.
One method: If I subtract equations $(1)$ and $(2)$  then I got: 
$$p\frac{\cos^4\theta}{\sin^2\theta} +q \frac{\cos^4\phi}{\sin^2\phi}=0$$
Please guide how to solve this.. thanks..
 [/INST] HINT:
$$p\sin\theta=q\sin\phi\implies p^2\sin^2\theta=q^2\sin^2\phi$$
$$\implies p^2\cos^2\theta-q^2\cos^2\phi=p^2-q^2\ \ \ \ (4)$$
Use $(2),(4)$ to solve for $\cos^2\phi,\cos^2\theta$
If $\cos^2\phi=y, \cot^2\phi=\frac{\cos^2\phi}{1-\cos^2\phi}=\frac y{1-y}$
Put the values of $\cot^2\phi,\cot^2\theta$ in $(1)$
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