2. Simplify the expression $\frac{2 \cos ^{2} x}{\sin x-\sin ^{-1} x}-\sin x$. For which values of $x$ does the expression have no meaning?

(6 points)
2. We consider $\sin ^{-1} x=\frac{1}{\sin x}$ and rearrange the denominator $\sin x-\sin ^{-1} x=\sin x-\frac{1}{\sin x}=\frac{\sin ^{2} x-1}{\sin x}$. We use the relationship between sine and cosine $\sin ^{2} x+\cos ^{2} x=1$, from which we express $\sin ^{2} x-1=-\cos ^{2} x$. We transform the fraction into $\mathrm{v} \frac{2 \cos ^{2} x}{\frac{-\cos 2 x}{\sin x}}=-2 \sin x, \cos x \neq 0$ and $\sin x \neq 0$ or $x \neq \frac{\pi}{2}+k \pi$ and $x \neq k \pi, k \in \mathbb{Z}$. We simplify the expression $-2 \sin x-\sin x=-3 \sin x$. The expression is undefined for $x=\frac{k \pi}{2}, k \in \mathbb{Z}$.

The simplified denominator $\sin x-\sin ^{-1} x=\sin x-\frac{1}{\sin x}=\frac{\sin ^{2} x-1}{\sin x} \ldots \ldots \ldots \ldots 1+1^{*}$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-13.jpg?height=88&width=1630&top_left_y=1755&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-13.jpg?height=54&width=1636&top_left_y=1829&top_left_x=210)

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-13.jpg?height=60&width=1627&top_left_y=1872&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-13.jpg?height=54&width=1625&top_left_y=1926&top_left_x=210)