Velvet Star Monitor

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How does $\frac{1}{2}\cosh(2x) -1 = \sinh^2(x)$?

Using hyperbolic trigonometric function identities is there a way to prove the following equation? $$\frac{1}{2} (\cosh(2x)-1) = \sinh^2(x)$$

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Vector Algebra Coordinate Transformation

Let us look at two coordinate systems $K$ and $K'$ with axes, respectively, $(x_1,x_2,x_3)$ and $(x_1',x_2',x_3')$ and unit vectors ($...

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Prove that lim of x/(x+1) = 1 as x approaches infinity

I want to prove that $$\lim\limits_{x\to \infty} \frac{x}{x+1}=1$$ I know that I need to show that: $$\left|\frac{1}{x+1}\right| \lt \epsilon$$ But I'...

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Why is 1 cubic meter 1000 liters?

Can anyone explain why $1\ \mathrm{m}^3$ is $1000$ liters? I just don't get it. 1 cubic meter is $1\times 1\times1$ meter. A cube. It has units $\mat...

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How to calculate the average molar mass of the atmosphere? [closed]

The task: Determine the average molar mass of the atmosphere - each for the moist air and the dry air. One example that I have calculated is this one: I h...

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Combinations with 10 digits

To find the number of $4$ digit combinations I can form with $10$ digits, without repeating any of the digits, I run the binomial coefficient of $10$ and ...

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Can someone explain ideals to me? [duplicate]

I'm struggling with the idea of ideals (both the definitions and the common notation). I'm in a basic collegiate algebra course, just looking fo...

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Example of a number that is normal in one base but not another

Wikipedia says that A real number, having a different base-$b$ expansion for each integer $b ≥ 2$, may be normal in one base but not in another. For bases...

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Probability concerning a 6-digit password

Consider the situation of decoding a 6-digit password that consists of the symbols $A$ to $Z$ and $0$ to $9$, where all possible combinations are tried ra...

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In Russian roulette, is it best to go first?

Assume that we are playing a game of Russian roulette (6 chambers) and that there is no shuffling after the shot is fired. I was wondering if you have an ...

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