Question$\cot x-\tan x=2 \cot 2 x$

Studdy Solution
Set the simplified left-hand side equal to the simplified right-hand side:
$\frac{\cos^2 x - \sin^2 x}{\sin x \cos x} = \frac{\cos^2 x - \sin^2 x}{\sin x \cos x}$
Since both sides are identical, the equation is satisfied for all $x$ where the expressions are defined. Therefore, the solutions are the values of $x$ that do not make the denominators zero:
$\sin x \cos x \neq 0$
This implies:
$x \neq n\pi \quad \text{and} \quad x \neq \frac{(2m+1)\pi}{2}, \quad \text{for integers } n \text{ and } m$
The solution is all $x$ except where $\sin x = 0$ or $\cos x = 0$, i.e., $x \neq n\pi$ and $x \neq \frac{(2m+1)\pi}{2}$.