How do you simplify $\mathrm{cos}\left(\mathrm{arctan}\left(x\right)\right)$ ?

Betsy Rhone
2021-12-15
Answered

How do you simplify $\mathrm{cos}\left(\mathrm{arctan}\left(x\right)\right)$ ?

You can still ask an expert for help

Melinda McCombs

Answered 2021-12-16
Author has **38** answers

Explanation:

Let simplify$\mathrm{cos}\left(\mathrm{arctan}\left(x\right)\right)$

Let$y=\mathrm{arctan}\left(x\right)$

$x=\mathrm{tan}\left(y\right)$

$x=\frac{\mathrm{sin}\left(y\right)}{\mathrm{cos}\left(y\right)}$

We need to have an expression for$\mathrm{cos}\left(y\right)$ only,

$x}^{2}=\frac{{\mathrm{sin}\left(y\right)}^{2}}{{\mathrm{cos}\left(y\right)}^{2}$

$x}^{2}+1=\frac{{\mathrm{sin}\left(y\right)}^{2}+{\mathrm{cos}\left(y\right)}^{2}}{{\mathrm{cos}\left(y\right)}^{2}$

$\frac{1}{{x}^{2}+1}={\mathrm{cos}\left(y\right)}^{2}$

$\frac{1}{\sqrt{{x}^{2}+1}}=\mathrm{cos}\left(y\right)=\mathrm{cos}\left(\mathrm{arctan}\left(x\right)\right)$

Let simplify

Let

We need to have an expression for

Buck Henry

Answered 2021-12-17
Author has **33** answers

Explanation:

Let$a=\mathrm{arctan}\left(x\right)$

The principal value of$a\in (-\frac{\pi}{2},\frac{\pi}{2})$

Then$\mathrm{tan}\alpha =x$ and

$0\le \mathrm{cos}\alpha \in [0,1)$ (wrongly marked as [0,-1], in my previous answer, two years ago).

Now the given expression is

$\mathrm{cos}\alpha =\frac{1}{\sqrt{1+{x}^{2}}},x\in (-\frac{\pi}{2},\frac{\pi}{2})$

It is important that$\mathrm{cos}\alpha \ge 0$ , for $\alpha \in {Q}_{1}$ or $Q}_{4$

If the piecewise-wholesome general inverse operator

$\left({\mathrm{tan}}^{-1}\right)$ is used

$\mathrm{cos}\left(\mathrm{tan}\right)}^{-1}x,=\pm \frac{1}{\sqrt{1+{x}^{2}}$

the negative sign is chosen, when$x\in {Q}_{3}$

Example:

$\mathrm{cos}\left(\mathrm{arctan}1\right)=\frac{1}{\sqrt{2}},\mathrm{arctan}1=\frac{\pi}{4}$

${\mathrm{cos}\left(\mathrm{tan}\right)}^{-1}=\mathrm{cos}(k\pi +\frac{\pi}{4},k=0,\pm 1,\pm 2,\pm ,\dots$

$\in {Q}_{1}$ or ${Q}_{3},x=\dots \frac{\pi}{4},\frac{5}{4}\pi ,\dots$

So, the value is$\pm \frac{1}{\sqrt{2}}$

My intention, in this approach, is to inform about nuances.

Let

The principal value of

Then

Now the given expression is

It is important that

If the piecewise-wholesome general inverse operator

the negative sign is chosen, when

Example:

So, the value is

My intention, in this approach, is to inform about nuances.

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