<s>[INST] How to simplify this trigonometric equation I am not sure how to further simplify this expression:
$$\sec^2(\arcsin(y/r)) \times \frac{\frac{1}{r}}{\sqrt{1-(\frac{y}{r})^2}} \times \frac{r}{1000}$$
How should I further simplify the $\sec^2(\arcsin(y/r))$?
The simplified version looks like this:
$$\left(\frac r{10\sqrt{r^2-y^2}}\right)^3$$
 [/INST] Starting from: $$\sec^2\left(\arcsin\left(\frac yr\right)\right) \times \frac{\frac{1}{r}}{\sqrt{1-(\frac{y}{r})^2}} \times \frac{r}{1000}.\tag 1$$
We know that $$\sec \left(\arcsin \left(x\right)\right)=\frac{\sqrt{1-x^2}}{1-x^2}\implies \sec^2 \left(\arcsin \left(x\right)\right)=\frac{1}{1-x^2}$$ then rewrite the $(1)$ like as:
$$\frac{1}{1-\left(\frac yr\right)^2}\times  \frac{\frac{1}{r}}{\sqrt{1-(\frac{y}{r})^2}} \times \frac{r}{1000} \tag 2$$
$$\frac{r^2}{r^2-y^2}\times \frac{\frac{1}{\not r}}{\sqrt{1-(\frac{y}{r})^2}}\times \frac{\not r}{1000}=\frac{r^2}{r^2-y^2}\times \frac{r}{\sqrt{r^2-y^2}}\times \frac{1}{10^3}=\left(\frac{r}{10\sqrt{r^2-y^2}}\right)^3$$
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