For questions regarding identities in algebraic structures, including the construction, composition, and interpretation thereof.

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8
votes
3answers
328 views

Proving the sum of squares of sine and cosine using the Cauchy product formula

Here are the power series of sine and cosine: $$\sin(x) = \sum_{n=0}^{\infty} (-1)^n \frac {x^{2n+1}} {(2n+1)!}$$ and $$\cos(x) = \sum_{n=0}^{\infty} (-1)^n \frac {x^{2n}} {(2n)!}$$ How can it be ...
2
votes
0answers
42 views

Is the product rule for logarithms an if-and-only-if statement?

If a function $f(x)$ is proportional to $\ln x$, then we know $$ f(xy) = f(x) + f(y). $$ My question is, Is the converse true? If we know that, for an unknown function f, $$ f(xy) = f(x) + f(y), $$ ...
5
votes
3answers
81 views

$\lim_{n \to \infty} \sqrt[n]{a^n+b^n}=\max \{a,b\}$

Let $a\geq 0$ and $ b\geq 0$. Prove that $\lim_{n \to \infty} \sqrt[n]{a^n+b^n}=\max \{a,b\}$. [Hint: Use the identity $(a^n -b^n)=(a-b)(\sum_{i=0}^{n-1}a^ib^{n-1-i})$] I need some help! I cannot ...
0
votes
1answer
37 views

$r+s \leq x+y$: How to prove it?

If the following were true, I could complete an exercise. Is it really true? If it is, has anybody some hint? If it is not, what the counter-example? Need some help! Thanks Let $t\in\Bbb{Q}$ and ...
1
vote
1answer
48 views

$a^n-b^n=(a-b) (\sum_{i=0}^{n-1}a^ib^{n-1-i})$ [duplicate]

How do I use finite induction to prove that $$a^n-b^n=(a-b) (\sum_{i=0}^{n-1}a^ib^{n-1-i}), \forall a,b\in \Bbb{R}\space \text{and} \space \forall n \in \Bbb{N}?$$ Ok, for $n=2$ it's fine. ...
0
votes
0answers
34 views

Simplifying a complex exponential equation

Can Someone please explain which identities are required to show that Thank you
-8
votes
6answers
242 views

How many mathematical identities that equal $1$ [closed]

I usually see some identities equal to $1$ , for examples $$\sin ^2(x)+\cos^2(x)=1$$ $$\sec ^2(x)-\tan^2(x)=1$$ $$\csc ^2(x)-\cot^2(x)=1$$ $$\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta ...
4
votes
4answers
157 views

Combinatoric proof for $\sum_{k=0}^n{n\choose k}\left(-1\right)^k\left(n-k\right)^4 = 0$ ($n\geqslant5$)

I'm trying to proove the following: $For\space every\space n \ge 5$: $$\sum_{k=0}^n{n\choose k}\left(-1\right)^k\left(n-k\right)^4 = 0$$ I've tried cancelling one $(n-k)$, and got this: ...
6
votes
1answer
56 views

An identity involving partial fractions decompositions

In Vladimir A. Smirnov's book Analytic Tools for Feynman Integrals (page 38), the following identity is suggested to perform partial fractions decompositions $$ \begin{split} ...
0
votes
2answers
34 views

Linear composition

can you help me with this quest? About composition $f$ and vector space $\mathbf{V}=\mathbb{Z^4_2}$ we know the following: $f \circ f = id_V$,$~~f $ $ \left(\begin{array}{ccc} 1\\ 0\\ 1\\ 0\\ ...
0
votes
0answers
26 views

Inner product identity

Let $u,v:\mathbb{R}^m\to\mathbb{R}^3$ be $C^1$ functions. I need to prove the following identity: $$ \langle \nabla u, \nabla v \rangle = \langle u,u \rangle \langle \nabla u, \nabla v \rangle - ...
2
votes
1answer
29 views

Finding the value of $y=b^2(3a^2+4ab+2b^2)$ if $a^2(2a^2+4ab+3b^2)=3$ and $a$ and $b$ are distinct zeros of $x^3-2x+c$

If $a$ and $b$ are distinct zeroes of the polynomial $x^3-2x+c$ and $$a^2(2a^2+4ab+3b^2)=3$$ $$b^2(3a^2+4ab+2b^2)=y$$ Evaluate $y$ I tried for many hours but couldn't solve this question. ...
12
votes
2answers
159 views

Solve $x^7-5x^4-x^3+4x+1=0$ for $x$

Solve for $x$ $$x^7-5x^4-x^3+4x+1=0$$ This equation has been bugging me since the past few days. I have found, using the Rational Root Theorem that $x=1$ is a root of this equation. However, ...
3
votes
0answers
32 views

Commutator $[A_{p,q},A_{s,t}]$ in the pure braid group?

Let $B_n$ be the braid group; that is, a group generated by $\sigma_1,\cdots,\sigma_{n-1}$ with relations $\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}$ for $i=1,\cdots,n-2$; ...
0
votes
1answer
15 views

Iterated sums identity

How to show that the following iterated sums are equal? $\sum_i \sum_j f(i)h(j)g(i,j) = \sum_j\bigg(\sum_i g(i,j)f(i)\bigg) h(j)$
2
votes
7answers
141 views

$f \circ g =\operatorname{ id}$ and $g \circ f \neq \operatorname{id}$?

Give two functions $f$ and $g$ s.t. $$f \circ g = \operatorname{id}$$ but $$g \circ f \neq \operatorname{id}$$ or a proof that this is impossible. This must be trivial, but I can't figure it out :) ...
20
votes
3answers
2k views

Find all the integral solutions to $x^6-y^6+3x^4y-3y^4x+y^3+3x^2+3x+1=0$

If $x,y\in\mathbb{Z}^+$, then find all the integral solutions to: $$x^6-y^6+3x^4y-3y^4x+y^3+3x^2+3x+1=0$$ I tried solving this question for an hour but still couldn't get it. I tried mod ...
0
votes
1answer
39 views

Proof for $∃xA⇔¬∀x¬A$

I want to prove, that $∃xA⇔¬∀x¬A$, using classic axioms. I think, I have to start with the following step: $∃xA⇔∃x¬¬A$ But I do not know, how to make this step, using axioms: $∃x¬¬A⇔¬∀x¬A$
-2
votes
1answer
44 views

Is this identity correct?

Is this identity true? Wolfram|Alpha thinks is not. $$x^{ln(x^3)} = e^{3\,[ln(x)]^2}$$ That's how I demonstrated it: $${\left(e^{ln(x)}\right)}^{3\,ln(x)} = e^{3\,[ln(x)]^2}$$ ...
0
votes
1answer
19 views

Does the following identity hold: $ [{A \times B^* + A^* \times B} ]$ = $2Re{[A \times B^*]}$

This seems to be true at first glance following that $a + a^*$ = $2Re(a)$ In any case, can someone help me verify whether this identity holds? Note: $\times$ is the cross product
0
votes
1answer
37 views

How to prove this identity (equality)?

I want to prove following equality: $$\begin{align*} &\left(\lambda_1 z_1^2+\lambda_2 z_2^2\right)\left(\frac{1}{\lambda_1} z_1^2 + \frac{1}{\lambda_2} z_2^2\right)=\\& =\frac{1}{4}\left ...
0
votes
1answer
41 views

Proof for $\forall x A \Leftrightarrow \neg \exists x \neg A$

I try to proof, that $\forall x A \Leftrightarrow \neg \exists x \neg A$ I know how to proof, that $\forall x A \Leftrightarrow \exists xA$, but I don't understand, how to get negation.
7
votes
1answer
82 views

Is there a proof of $\sum_{n=0}^x {{(-1)^n(x-n+1)^x}\over{n!(x-n)!}} = 1$ using induction?

Can someone prove (or disprove) this equality? $$\sum_{n=0}^x {{(-1)^n(x-n+1)^x}\over{n!(x-n)!}} = 1$$ where the value of $x$ can vary. This is a pattern found in derivatives and stuff but I'm not ...
1
vote
0answers
41 views

New identity for sums of Bessel functions?

I've come across the following proposed identity: $$ ...
2
votes
1answer
64 views

Are $\sin(\alpha\beta)$ and $\sin(\alpha^{\beta})$ expressible in terms of $\sin(\alpha)$ and $\sin(\beta)$?

There is a well known formula for expressing $\sin(\alpha+\beta)$ just using $\sin(\alpha)$ and $\sin(\beta)$. It is enough to replace $\cos$ in the formula ...
1
vote
1answer
30 views

Determine integral by using the following identity (which is imaginairy)?

I want to determine the following integral: $$\int_{-\infty}^\infty \frac1{x^6+1} dx$$ by using the following identity: $$\frac1{x^6+1} = \Im\left[\frac1{x^3-i}\right]$$ How in the world can I do ...
4
votes
1answer
35 views

Geometric interpretation of linear forms in the sum of four (or eight) squares identity

There is a well-known sum-of-squares identity $$(a^2+b^2)(c^2+d^2)=(ac+bd)^2+(ad-bc)^2. \tag{1}$$ Moreover, letting $\vec{u}=[\begin{smallmatrix}a\\b\end{smallmatrix}]$, ...
1
vote
1answer
32 views

Which identity is being used to get $\sin(wa)\cos(wt)=\sin(w(a+t))+\sin(w(a-t))$?

Which identity is being used to get $\sin(wa)\cos(wt)=\frac{\sin(w(a+t))+\sin(w(a-t))}{2}$? Couldn't find it among the trigonometric identities.
0
votes
4answers
40 views

Help in proving an algebraic identity involving powers of binomials.

For some reason I found this equation: $(1 + x)^n - 1 = x \sum\limits_{k=0}^{n-1} (1+x)^k$ I think that this is an identity. If for instance one expands the powers and the sum for n = 4, the ...
0
votes
1answer
24 views

Prove this trig identity?

I'm having trouble proving the following identity: $$A\cos(\omega t) + B\sin(\omega t) = \sqrt{A^2 + B^2} \cos(\omega t - \arctan(\tfrac{B}{A}))$$ Does anyone know how this can be done? Thanks!
1
vote
2answers
41 views

Two partial fraction identities for $\frac{x^n}{x^m+k}$

Consider the following expression: $$\frac{x^n}{x^m+k},$$ for non-negative integers $n$ and $m$, $m>n$, and $k\in\mathbb{C}$. For $k=0$ the expression clearly simplifies to $x^{n-m}$. For ...
3
votes
0answers
40 views

Decide if radical expression equals a given rational number

Given a radical expression an a rational number. Is there an algorithm to decide if the expression equals the number? Example: $(\sqrt{\sqrt[5]{74} - \sqrt[14]{78}})^{356}+\sqrt[6]{63} \overset{?}= ...
0
votes
2answers
37 views

Which identity has been used here?

I have this written down in my notes, but I cannot remember how it came about: $$\sin(3t)\cos(10t) = 0.5(\sin (13t) + \sin (-7t))$$
0
votes
3answers
50 views

How is $A\sin\theta +B\cos\theta = C\sin(\theta + \phi)$ derived?

I have come across this trig identity and I want to understand how it was derived. I have never seen it before, nor have I seen it in any of the online resources including the many trig identity cheat ...
3
votes
2answers
83 views

Tough trigonometric identity

Prove that $$\cot 13^o\cot 23^o \tan 31^o\tan35^o\tan41^o = \tan 75^o$$ I managed to rearrange it to the form $$\tan 31^o\tan 35^o\cot 49^o = \cot 15^o\tan 23^o\cot 77^o$$ and in this form we have ...
0
votes
0answers
126 views

Why is Wolfram Alpha wrong?

I calculated $$\tan 75^o - [\cos 13^o\cdot \cot 23^o \cdot \tan 31^o \cdot \tan 35^o\cdot \tan41^o]$$ and I got a nonzero answer: ...
1
vote
1answer
37 views

Prove $ \nabla \cdot (f \nabla \psi ) = \nabla f \cdot \nabla \psi + f \nabla ^2 \psi $ in general curvilinear coordinates.

Prove $ \nabla \cdot (f \nabla \psi ) = \nabla f \cdot \nabla \psi + f \nabla ^2 \psi $ in general curvilinear coordinates. I have been attempting to do this using general curvilinear dot products ...
27
votes
6answers
3k views

What Gauss *could* have meant?

I was reading the Wikipedia entry on Euler's identity ($e^{i\pi}+1=0$) and I came across this statement: "The mathematician Carl Friedrich Gauss was reported to have commented that if this formula ...
3
votes
2answers
42 views

Algebraic identity I cannot solve

$$ \sum_{i<j}\sum_{j=1}^N(y_i-y_j)^2= N \sum_{j=1}^N(y_i-\bar{y})^2 $$ Where, $\bar{y}$ is the average. This is what I did: $$ ...
1
vote
1answer
34 views

Number Systems: Determining when they have closure, identities, inverses, and more.

I have the following $9$ number systems at hand and I am to determine which of them possess a particular property. I am having trouble understanding some of the subtleties between the questions and ...
5
votes
3answers
226 views

Using an identity to simplify the sum

So I ran into this problem today. It asks me to use an identity to simplify the sum. $$\sum_{j=7}^{27}\ln\left(\frac{j+1}{j}\right)$$ I have no idea where to start. I don't know any ...
2
votes
1answer
64 views

Can $2a^2+2a+2ab^2+b^2$ be written algebraically as the sum of three triangular numbers?

Let $T(n)=\tfrac{1}{2}n(n+1)$ denote the $n$th triangular number. I'm looking for an identity of the form $$ 2a^2+2a+2ab^2+b^2 = T(f(a,b)) + T(g(a,b)) + T(h(a,b))\tag{$\star$} $$ where $a,b$ are ...
0
votes
1answer
24 views

Vector calculus identities

Let $f$ be scalar potential for the vector field $\underline u $ (i.e $\underline u = -\underline \nabla f$). Prove that the vector field $$ \underline r \wedge \underline u $$ has magnetic ...
1
vote
2answers
39 views

Identity involving trigonometric sum

I have to prove that $$\overset{N}{\underset{n=-N}{\sum}} \left(N-\left|n\right|\right)e^{2\pi inx}=\left|\overset{N}{\underset{n=1}{\sum}} e^{2\pi inx}\right|^{2}=\left(\frac{\sin\left(N\pi ...
24
votes
2answers
346 views

Disproving an “almost true” trigonometric identity

The plausible looking "identity" $$\sin(\frac{\pi}{51})+\cos(\frac{\pi}{74})=\frac{3}{2\sqrt 2}$$ is not true, but it is close indeed: $$LHS=1.0606\color{blue}{598...}$$ ...
0
votes
1answer
86 views

What trig. identity would help solve $2 + \cos(2x) = 3\cos(x)$?

I need help with a homework question that has me puzzled. I need to solve the following equation: $$2 + \cos(2x) = 3\cos(x)$$ I don't see a good trig identity to apply. I tried $\cos(2x) = ...
0
votes
1answer
45 views

Prove that $\sum_{k=1}^{\frac{n-1}{2}}\cos\left(\frac{2\pi k}{n}\right)=-\frac{1}{2}$ if $n=1\mod 2$

I found out that this equality holds by accident,$$\sum_{k=1}^{\frac{n-1}{2}}\cos\left(\frac{2\pi k}{n}\right)=-\frac{1}{2}$$ if $n=1\mod 2$. However, I am not able to prove this directly with rules ...
2
votes
2answers
95 views

Product rule for simplex numbers

The $n$th triangular number is defined as $T_2(n) = n(n+1)/2$, and there is an interesting product rule for triangular numbers: $$T_2(mn) = T_2(m)\,T_2(n) + T_2(m-1)\,T_2(n-1).$$ The tetrahedral ...
0
votes
1answer
106 views

Polynomial identities

When I was about 17 our teacher showed us how polynomial identities had equal coefficients. I remember him showing that this was so by moving one polynomial "over to the other side" and showing that ...
1
vote
1answer
49 views

Proving $\sum_{k=0}^n\dfrac{x_k^{n+1}}{\prod_{j\neq k}(x_k-x_j)}=\sum_{k=0}^nx_k$

In Problems from the book by Andreescu, there's the following problem : Let $x_0,\ldots,x_n$ be distinct complex numbers. Prove $\displaystyle \sum_{k=0}^n\dfrac{x_k^{n+1}}{\prod_{j\neq ...