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4
votes
1answer
21 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
28 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
26 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
22 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
38 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
31 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
36 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
44 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
77 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
111 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
30 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 ...
23
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
39 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
27 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
217 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
59 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
22 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 ...
0
votes
2answers
34 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 ...
22
votes
2answers
308 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
53 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
44 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
80 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
100 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
47 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 ...
1
vote
0answers
54 views

Generalization of trace norm identity

Given a $2\times 2$ complex matrix $M$, the sum of its singular values (i.e. the trace norm) can be written as: $$\mathrm{Tr}\,|M|=\sqrt{\mathrm{Tr}(M^\dagger M)+2|\mathrm{Det}(M)|}$$ Is anyone aware ...
4
votes
1answer
62 views

Are there more examples of functional equations which are also valid for the identity map?

I find the co-incidence of the identity: $$\sin(A+B)\sin(A-B) = \sin^2 A - \sin^2 B$$ very pleasing. So, I was wondering if there are more of these type of identities. To make my question precise: ...
3
votes
2answers
60 views

Find the Value of Trigonometric Expression

Given $$\begin{align} \frac{\cos \alpha}{\cos \beta}+\frac{\sin \alpha}{\sin \beta}=-1 \end{align} \tag{1}$$ Find the value of $$\begin{align} \frac{\cos^3 \beta}{\cos \alpha}+\frac{\sin ...
-2
votes
2answers
98 views

find a formula for $\sin3 \theta$ in terms of $\sin \theta$,$\cos \theta$ by using an angle-sum formula for sin(2θ+θ) [closed]

Find a formula for $\sin3 \theta$ in terms of $\sin \theta$,$\cos \theta$ by using an angle-sum formula for $\sin(2 \theta+\theta$)
2
votes
3answers
159 views

Deriving the sum-to-product identities

I've been asked by my textbook to derive the "sum-to-product" identities from the "product-to-sum" identities. I've attempted to to do this but i've met a dead end, and i'm quite confused. Using ...
0
votes
0answers
40 views

Proof of Lehmer's Identity

Is there any elementary proof for Lehmer's Identity ? let $n$ be a positive integer. Prove that $$\sum_{k=0}^{n}\binom{6n+5}{6k+2}B_{6k+2}=\frac{6n+5}{3}$$ where $B_n$ denotes the $n$th Bernoulli ...
3
votes
2answers
53 views

Find the value of $\frac{S_{5}S_{2}}{S_{7}}$

If $a$, $b$, $c$ $\in \mathbb R$, we define $S_{k}=\frac{a^k+b^k+c^k}{k}$ (where $k$ is a non-negative integer). Given that $S_{1}=0$, find the value of $$\frac{S_{5}S_{2}}{S_{7}}$$ I tried: ...
1
vote
0answers
45 views

Sum involving binomial coefficients

Exist a closed form for $$\left(-1\right)^{N}\underset{i=1}{\overset{N}{\sum}}\left(-1\right)^{i}\dbinom{N}{i}\dbinom{N+i}{i-1}\,\frac{1}{2i+1}?$$ I think I've to use in some way the formula of the ...
0
votes
2answers
53 views

Prove trigonometry identity for $\sin A+\cos A$

I’ve been struggling in proving this identity for hours (yes, shame on me), but I can’t see any light. $\frac { \cos(A) }{ 1-\tan(A) } +\frac { \sin(A) }{ 1-\cot(A) } =\sin(A)+\cos(A)$ I've been ...
2
votes
1answer
36 views

What is Vandermonde's formula with multisets?

I need Vandermonde's formula in multi-set form. I modified the original formula but I get a mess with too many letters everywhere, is there a nice representation? Here's the original: $$ ...
0
votes
1answer
17 views

How to demonstrate a vector norm identity?

I'm trying to demonstrate the following identity: $|| y-x ||^{2}= ||\frac{1}{2}(x+y) \, ||^{2}- ||\frac{1}{2}(x-y) \, ||^{2}+i||\frac{1}{2}(x+iy) \, ||^{2}- i||\frac{1}{2}(x-iy) \, ||^2$ I've tried ...
5
votes
1answer
124 views

Prove that $512^3 + 675^3 + 720^3$ is a composite number

We have to prove that the number $$N=512^3 + 675^3 + 720^3$$ is composite. I tried to use the identity $(a^3+b^3+c^3)=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)+3abc$ hoping to take out some common ...
0
votes
3answers
61 views

applications of the identity $ab + \left(\frac{a+b}{2} - b\right)^2 = \left(\frac{a+b}{2}\right)^2$

I am reading euclid's elements I found the algebraic identity $ab + \left(\frac{a+b}{2} - b\right)^2 = \left(\frac{a+b}{2}\right)^2$ I ponder on usage of this identity for $2$ hours. but I can't ...
6
votes
0answers
101 views

Specializations of elementary symmetric polynomials

Let $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$ be a set of $n$ indeterminates. The $h^{th}$elementary symmetric polynomial is the sum of all monomials with $h$ factors \begin{eqnarray*} ...
0
votes
1answer
51 views

Trigonometric Functions using factorisation and small angle identities

The function f(x)=sin3x - sin2x + sins is defined for the domain 0 $\le$ x $\le$ $\frac{\pi}{2}$. a) By method of factorisation, show that f(x) = sin2x(2cosx - 1). b) Hence solve the equation f(x) ...
3
votes
0answers
54 views

Cube root of two continued fraction

I know there is a nice way of getting the continued fraction expansion of quadratic irrationals mainly because they recur after a point, and if they recur after a point they are quadratic irrationals. ...
0
votes
0answers
25 views

Prove algebraic identity

Let $j\ge 0$ be an integer and $0<k<t_c$ and $\omega_0,\beta$ be real numbers. The question is to prove the following identity: \begin{eqnarray} j! \sum\limits_{s=\pm} \left(\begin{array}{c} ...
2
votes
1answer
25 views

Matrices derivation and identities

Good day, I am having difficulty understanding the derivation below. This is adopted from Simon Prince's computer vision book, pg 543 for the derivation, pg 626 for the inversion relation. I can not ...
2
votes
3answers
56 views

Finding tan(A+B)

So I know that $$ \tan(A+B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A) \tan(B)}, $$ but I don`t know how to find $\tan(B)$ for the following problem: If $\tan A = 2/3$ and $\sin B = 5/\sqrt{41}$ and ...
0
votes
2answers
25 views

Polynomial identity for a sum

If $$f(x) = \sum_{i=0}^{n}A_i x^i \quad \text{ and } \quad g(x) = \sum_{i=0}^{n}B_i x^i$$ are two degree $n$ polynomials, then we can say that the polynomial $$h(x) = \sum_{k=0}^{2n}C_k x^k \quad ...
0
votes
0answers
19 views

Identity for fractional summation

I would like to know if there's an identity to represent the following summation $\sum_{i=0}^{n}\frac{x_i}{y_i}$ Where x and y are non integer values. The result of this is being calculated using ...
2
votes
1answer
81 views

Do we lose everything, if the natural transformations in a monad are exactly inverse?

I'm currently explaining monads $$T:{\bf C}\to{\bf C},\hspace{1cm}\eta:1_{\bf C}\to T,\hspace{1cm}\mu:T\circ T\to T,$$ to my brain and the "only" tricky thing are really the identity relations. I ...
3
votes
2answers
93 views

Manipulating identities

I'm having some trouble deriving certain identities. If $$S(z) = \prod_{i=1}^n (z-z_i)$$ then how can I write $$\frac{1}{S(z)}\frac{d^2S}{dz^2} = \sum_{i=1}^n\frac{1}{z-z_i}\sum_{j\neq ...
5
votes
2answers
115 views

What does the Yoneda lemma say for the identity functor and finite sets?

So I try to plug in the simplest arguments into the Yoneda lemma and see how to interpret it. I'll try it for the identity functor and the category of finite sets, in particular, I use an three ...
2
votes
0answers
57 views

A questions about Group Rings

Let's say $R:=\mathbb{Z}_p[C_{p^\infty}]$ be the group ring of a Prufer group over the field of integer module a prime $p$. We have $C_{p^\infty}=\langle u_1, u_2, ..., u_n, ... |\,\,\,\, ...
2
votes
3answers
43 views

An identity in Ring of characteristic $p$ prime

Is it true that in a ring of prime characteristic $p$ results that $(x-1)^{p-1}=1+x+x^2+...+x^{p-1}$ ? If this is not true in general, the assumption that $x$ is a nilpotent element (let's say ...