# Is there a way to isolate x for tanh x = 1?

• silicon_hobo
In summary, the conversation discusses a solution for isolating x when tanh x = 1 and also mentions the values of cosh and sinh in terms of exponentials. The conversation also includes a discussion on using trig identities to show that x = 0 when d/dx tanh x = 1. The solution ultimately involves setting tanh x = 0 and using a trig identity to show that x = 0 is the only solution.
silicon_hobo
[SOLVED] is there a way to isolate x?

## Homework Statement

Where does the tangent of tanh x = 1?

## Homework Equations

$$\\f(x)=\tanh x$$
$$f^\prime(x)=\\sech^2x=1-\tanh^2x$$

## The Attempt at a Solution

$$\\f^\prime(x)=1-\tanh^2x\rightarrow1-\tanh^2x=1\rightarrow\sqrt{\tanh^2x}=\sqrt{0}\rightarrow \tanh x=0$$

Since I have shown that the tangent = 1 when tanh x = 0, I thought it may be sufficient to simply add another line saying x = 0 since we know (based on the definition of tanh) that if tanh x = 0 then x = 0. However, I am wondering if there is a way to approach this to actually isolate x without using this assumption? Does that make sense? Thanks for reading.

Last edited:
what does cosh and sinh equal in terms of exponentials?

You can...

as you said, $$\\sech^2x = 1 - \\tanh^2x$$
so for $$\\sech^2x = 1$$ then
$$1- \frac{e^x-e^{-x}}{e^x+e^{-x}} = 1$$

$$\Rightarrow$$ ...

Last edited:
You know that tan(alpha) = 1 when alpha = pi/4. So now just write tanhx in terms of exponentials and solve using simple algebra (tanhx = pi/4).

Alright, I think I've got it. Continuing from where I left off:

$$0 = \tanh x =\frac{e^{2x}-1}{e^{2x}+1}$$

Therefore, $$e^{2x}-1=0$$ which gives $$x=0$$ as the only solution. Looks good yeah? Cheers.

why are you setting tanhx = 0?...

So I could, in turn, use a trig identity to show that x = 0 when d/dx tanh x = 1. It should be correct if you read from the beginning. Would you happen to know how to mark posts as [SOLVED] do you? Cheers.

## 1. Is it possible to isolate x?

Yes, it is possible to isolate x through various methods such as chromatography, distillation, and extraction.

## 2. What is the purpose of isolating x?

The purpose of isolating x is to separate it from other components or substances in a mixture for further analysis or use.

## 3. How do you determine the purity of isolated x?

The purity of isolated x can be determined through techniques such as melting point analysis, spectroscopy, and titration.

## 4. Can x be isolated from any type of sample?

It depends on the properties of x and the nature of the sample. Some substances may be more difficult to isolate than others.

## 5. What are the potential challenges in isolating x?

The main challenges in isolating x are determining the most suitable method for isolation and ensuring the purity of the isolated substance. Other factors such as cost, time, and availability of equipment and resources may also pose challenges.

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