Question

Find at least one solution to the following equation:
(sin(x^2 − 1))/(1 − sin(x^2 − 1)) = sin(x) + sin^2(x) + sin^3(x) + sin^4(x) + · · ·

Answer 1

Let's suppose $\sin(x) \neq \pm1, \sin(x^2-1)\neq1$ (as if it is not the case either the right side or the left side diverges). In this case the expression on the right converges to :

$\sum_{k\geq 1} \sin^k(x) = \frac{\sin(x)}{1-\sin(x)}$ by the geometric progression sum formula.

Therefore we have :

$\frac{\sin(x^2-1)}{1-\sin(x^2-1)}=\frac{\sin(x)}{1-\sin(x)}$

As we need to find at least one solution, we will not seek the general solution of this equation and we will study at least the case :

$x^2-1=x$ (as the solutions of this equations are also solutions of our equation)

$x_{+,-}=\frac{1\pm\sqrt{1+4}}{2} = \frac{1}{2}\pm \frac{\sqrt{5}}{2}$

We know that both $x_{+,-}\neq \pm \frac{\pi}{2}$ , so this solution satisfies our condition $\sin(x),\sin(x^2-1) \neq \pm 1$. And thus we have found at least one (even two) solutions.