1. **State the problem:** Solve or analyze the equation $$x^2 - y^2 = \tan(xy)$$ where $x$ and $y$ are variables. 2. **Understand the equation:** This is a transcendental equation involving polynomial terms $x^2$, $y^2$ and the trigonometric function $\tan(xy)$. 3. **Rewrite the equation:** $$x^2 - y^2 = \tan(xy)$$ 4. **Consider special cases:** - If $xy = k\pi$ for integer $k$, then $\tan(xy) = 0$, so the equation reduces to: $$x^2 - y^2 = 0 \implies x^2 = y^2 \implies y = \pm x$$ - If $x=0$ or $y=0$, then $\tan(0) = 0$, so: $$x^2 - y^2 = 0$$ which again implies $y = \pm x$. 5. **General approach:** Because $\tan$ is periodic and nonlinear, explicit closed-form solutions for $y$ in terms of $x$ are not generally possible. 6. **Numerical or graphical methods:** To find solutions for specific $x$, one can numerically solve: $$x^2 - y^2 - \tan(xy) = 0$$ for $y$. 7. **Summary:** - The equation mixes polynomial and transcendental terms. - Solutions include $y = \pm x$ when $xy = k\pi$. - Other solutions require numerical methods. **Final answer:** The equation defines implicit relations between $x$ and $y$ with solutions at $y=\pm x$ for $xy = k\pi$, and other solutions found numerically.