Given $$p\sin^{4}{\theta}-q\sin^{4}{\phi}=p$$ and $$p\cos^{4}{\theta}-q\cos^{4}{\phi}=q$$ find $\theta$ and $\phi$.
Here is my solution (help improving it would be much appreciated):
$$p(1-2\cos^{2}{\theta}+\cos^{4}{\theta})-q(1-2\cos^{2}{\phi}+\cos^{4}{\phi})=p \tag1$$
Subtracting this from the original second equation gives $$p-2p\cos^{2}{\theta}-q+2q\cos^{2}{\phi}=p-q \tag2$$
Rearranging: $$\cos^{2}{\phi}=\frac{p}{q}\cos^{2}{\theta} \tag3$$
Which means $$\cos^{4}{\phi}=\frac{p^{2}}{q^{2}}\cos^{4}{\theta} \tag4$$
When substituted into the second original equation we get $$\left(p-\frac{p^{2}}{q}\right)\cos^{4}{\theta}=q \tag5$$
From this, $$\cos^{4}{\theta}=\frac{q^{2}}{qp-p^{2}} \tag6$$ and $$\cos^{4}{\phi}=\frac{p^{2}}{qp-p^{2}} \tag7$$
BUT... is it possible to obtain "nicer" expressions for $\theta$ and $\phi$ somehow?