\section*{Problem 5 - 091245}

All real numbers \(\lambda\) are to be determined for which the equation

\[
\sin ^{4} x-\cos ^{4} x=\lambda\left(\tan ^{4} x-\cot ^{4} x\right)
\]

a) has no, b) has exactly one, c) has exactly two, d) has more than two real solutions in the interval \(0<x<\frac{\pi}{2}\).
}

First, it is clear that the initial equation

\[
\sin ^{4} x-\cos ^{4} x=\lambda\left(\tan ^{4} x-\cot ^{4} x\right)
\]

always has the solution \(x=\frac{\pi}{4}\) for any \(\lambda \in \mathbb{R}\).

Furthermore, every solution \(\tilde{x} \in\left(0, \frac{\pi}{4}\right)\) is uniquely and reversibly associated with the solution \(\frac{\pi}{2}-\tilde{x}\), since \(\left(^{*}\right)\) is obviously invariant under the exchange \(x \leftrightarrow \frac{\pi}{2}-x\).

From this preliminary remark, it already follows that the cases a) and c) in the problem statement cannot occur for a given \(\lambda \in \mathbb{R}\). Moreover, we can restrict our search for further solutions other than \(x=\frac{\pi}{4}\) to the interval \((0, \frac{\pi}{4})\) for a given \(\lambda \in \mathbb{R}\), which concerns cases b) and d).

To this end, we consider the function \(\tilde{\lambda}:\left(0, \frac{\pi}{4}\right) \rightarrow \mathbb{R}\) defined by

\[
\tilde{\lambda}(x)=\frac{\sin ^{4} x-\cos ^{4} x}{\tan ^{4} x-\cot ^{4} x}
\]

which, after simplifying by \(\sin ^{4} x-\cos ^{4} x \neq 0\), can also be written more simply as

\[
\tilde{\lambda}(x)=\frac{\sin ^{4} x \cos ^{4} x}{\sin ^{4} x+\cos ^{4} x}=\frac{\sin ^{4}(2 x)}{8\left(2-\sin ^{2}(2 x)\right)}
\]

From the second representation, it follows immediately that \(\tilde{\lambda}(x)\) is strictly increasing on \(\left(0, \frac{\pi}{4}\right)\) and takes every value in \(\left(0, \frac{1}{8}\right)\) exactly once.

Case d) (with exactly 3 solutions) thus occurs precisely for \(\lambda \in\left(0, \frac{1}{8}\right)\), while for the other \(\lambda \in \mathbb{R} \backslash\left(0, \frac{1}{8}\right)\), there remains only the one solution \(x=\frac{\pi}{4}\), which corresponds to case b).