From the following two linear homogeneous algebraic equations:
$$A \sin\left(\frac{kl}{\sqrt2}\right) = B \sin(kl)$$ $$\frac{kA}{\sqrt2}\cos\left(\frac{kl}{\sqrt2}\right) = kB\cos(kl)$$
form matrix of these 2 equations, and setting the determinant equal to zero will lead to: $$\frac1{\sqrt2}\cos\left(\frac{kl}{\sqrt2}\right)\sin(kl) - \sin\left(\frac{kl}{\sqrt2}\right)\cos(kl) = 0.$$
k is unknown. l is constant. A and B are constants. I'm trying to find a nonzero solution when the determinant of the equation system vanishes. How do I solve this?