Intereting Posts

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If $G$ contains a normal subgroup $H \cong \mathbb{Z_2}$ such that $G/H$ is infinite cyclic, then $G \cong \mathbb{Z} \times \mathbb{Z_2}$

Verify the identity: $\tan^{-1} x + \tan^{-1} (1/x) = \frac\pi 2, x > 0$

$$\alpha= \tan^{-1} x$$

$$\beta = \tan^{-1} (1/x)$$

- Divergence of $\sum\frac{\cos(\sqrt{n}x)}{\sqrt{n}}$
- A little integration paradox
- Help needed with trigonometric identity
- what is sine of a real number
- How to find an end point of an arc given another end point, radius, and arc direction?
- Why does Arccos(Sin(x)) look like this??

$$\tan \alpha = x$$

$$\tan \beta = 1/x$$

$$\tan^{-1}[\tan(\alpha + \beta)]$$

$$\tan^{-1}\left

[{\tan\alpha + \tan\beta\over 1 – \tan\alpha \tan\beta}

\right]$$

$$\tan^{-1}\left[

{x + 1/x\over 1- x/x }\right]$$

$$\tan^{-1}\left[{x + (1/x)\over 0} \right]$$

I can’t find out what I’m doing wrong..

- Trigonometric Equation $\sin x=\tan\frac{\pi}{15}\tan\frac{4\pi}{15}\tan\frac{3\pi}{10}\tan\frac{6\pi}{15}$
- A triangle determinant that is always zero
- How to find the exact value of $ \cos(36^\circ) $?
- ArcTan(2) a rational multiple of $\pi$?
- Find the coordinates of a point on a circle
- Find length of $CD$ where $\measuredangle BCA=120^\circ$ and $CD$ is the bisector of $\measuredangle BCA$ meeting $AB$ at $D$
- Proving a relation between inradius ,circumradius and exradii in a triangle
- Evaluating a double integral involving exponential of trigonometric functions
- Equilateral Triangle Problem With Trig
- How would you find the trigonometric roots of a cubic?

**Hint:**

When you want to prove that something smooth is constant, use derivatives.

details:

if $f(x) = \arctan x + \arctan\frac 1x$ then

$$

f'(x) = \frac 1{1+x^2} + \frac 1{1+\left(\frac 1x\right)^2}\times \left(-\frac{1}{x^2}\right) =0

$$

then $f(x) = f(1) = 2\arctan 1 = \frac\pi 2$ on the **interval** $\{x>0\}$.

The problem of your method is that the formula you are using is true only when

$$

\alpha , \beta, \alpha + \beta \neq \frac\pi 2 \mod \pi

$$

An easy, mostly graphical proof: $\tan\alpha=x$, $\tan\beta=\frac1x$, and $\alpha+\beta=\frac\pi2$.

The reason you get a division by zero in the argument of arctan is that $\displaystyle\lim_{\varphi\to\frac\pi2}\tan\varphi=\pm\infty\approx\tfrac10$. So, in ** very** informal notation, you could say that $\tan^{-1}(\infty)=\tfrac\pi2$, and that your calculation in a way make sense.

You’re basically trying to compute $\tan(\pi/2)$, which doesn’t exist.

If you set $\beta=\arctan(1/x)$, then $\tan\beta=1/x$, that is

$$

x=\cot\beta=\tan\left(\frac{\pi}{2}-\beta\right)

$$

Therefore

$$

\arctan x=\arctan\tan\left(\frac{\pi}{2}-\beta\right)=\frac{\pi}{2}-\beta

$$

by the hypothesis that $x>0$, so that $0<\arctan(1/x)<\pi/2$.

One may also use complex numbers: We are multiplying two complex numbers with argument $\frac{1}{x}$ and $x$.

So, we desire to show that $\arg((1 + ix)(x + i)) = \frac{\pi}{2}$

We expand the product to get $(x^2 + 1)i$ — since there is no real part and the imaginary part is $> 0$, the argument is $\frac{\pi}{2}$

Assume $x>0$, then

\begin{align}&\tan^{-1} x +\tan^{-1} \dfrac1x

\\\\=&\tan^{-1} x +\tan^{-1}\dfrac1{\tan\tan^{-1} x}

\\\\=&\tan^{-1} x +\tan^{-1}\cot \tan^{-1} x

\\\\=&\tan^{-1} x +\tan^{-1}\tan(\dfrac{\pi}2- \tan^{-1}x)

\\\\=&\tan^{-1} x +\dfrac{\pi}2- \tan^{-1}x\qquad\qquad\qquad

\left(\because\text{for $x>0$,}\;\dfrac{\pi}2- \tan^{-1}x\in\left(0,\dfrac{\pi}2\right)\right)

\\\\=&\dfrac{\pi}2.

\end{align}

Yet another possibility:

You want $y+z$ for $y,z$ satisfying $\sin y / \cos y=\cos z/\sin z$. Since $\sin y=\cos(\pi/2-y)$ it follows that $z=\pi/2-y$.

- Differentiating $y=x^{2}$
- Does the following series converge?
- Is $\left(\sum\limits_{k=1}^\infty \frac{x_k}{j+k}\right)_{j\geq 1}\in\ell_2$ true if $(x_k)_{k\geq 1}\in\ell_2$
- $\mathcal{F_t}$-martingales with Itô's formula?
- Another integral with Catalan
- For any convex polygon there is a line that divides both its area and perimeter in half.
- Does there exist a complex function which is differentiable at one point and nowhere else continuous?
- Base conversion: How to convert between Decimal and a Complex base?
- Rudin Theorem 3.27
- Uniqueness in the Riesz representation theorem for the dual of $C(X)$ in the book by Royden
- Find the area of largest rectangle that can be inscribed in an ellipse
- What is the significance of reversing the polarity of the negative eigenvalues of a symmetric matrix?
- Seeking a more direct proof for: $m+n\mid f(m)+f(n)\implies m-n\mid f(m)-f(n)$
- Can there be two distinct, continuous functions that are equal at all rationals?
- Prove the ring $a+b\sqrt{2}+c\sqrt{4}$ has inverse and is a field