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

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0
votes
1answer
25 views

How to show a norm identity of a weighted sum

I ran across the following identity while reading up on norms. It deals with the square of the $2$-norm of a convex combination. That is, for all $x,y,\in\mathbb{R}^{n}$ and $\rho \in [0,1]$: ...
2
votes
2answers
43 views

Cosine and Sine of Sums

What's a good way to simplify $\sin( \sum\nolimits_{i=1}^{\infty} x_i)$ as the product and sum of $\sin(x_i)$ and $\cos(x_i)$ alone? And the same for $\cos( \sum\nolimits_{i=1}^{\infty} x_i)$?
17
votes
6answers
387 views

Product of cosines: $ \prod_{r=1}^{7} \cos \frac{r\pi}{15} $

Evaluate $$ \prod_{r=1}^{7} \cos {\dfrac{r\pi}{15}} $$ I tried trigonometric identities of product of cosines, i.e, $$\cos\text{A}\cdot\cos\text{B} = \dfrac{1}{2}[ \cos(A+B)+\cos(A-B)] ...
1
vote
1answer
42 views

Is that hyperbolic identity correct?

Given the expression: \begin{equation} |x|\cosh(kx)+x\sinh(kx), \;\; k>0 \end{equation} By taking cases for $x$, we have: \begin{equation} \bullet \quad x>0: x(\cosh(kx)+\sinh(kx))=x\left( ...
-3
votes
3answers
72 views

How to prove $(\tan a + \tan b) / (\tan ab) = (\tan ab) / (\tan a - \tan b)$? [closed]

How would I prove $\dfrac{\tan a + \tan b }{ \tan ab }=\dfrac{ \tan ab } {\tan a - \tan b}$? I have no idea because I've exhausted all possibilities that I know of, I've tried multiplying by the ...
2
votes
6answers
97 views

Alternative proof of $\cos^6{\theta}+\sin^6{\theta}=\frac{1}{8}(5+3\cos{4\theta})$

In this question, the only proof of the trigonometric identity: $$\cos^6{\theta}+\sin^6{\theta}=\frac{1}{8}(5+3\cos{4\theta})$$ is via factoring the sum of cubes: ...
0
votes
0answers
35 views

Identification of two finite fields

I got the assignment to determine the identification of $\mathbb{F}_3^2$ and $\mathbb{F}_{3^2}$. I am capable of constructing the non-prime fields $\mathbb{F}_{3^2}$ as a reduction of $\mathbb{F}_3$ ...
5
votes
0answers
62 views

showing that an inequality holds

I am trying to figure out how to show that for $n\geq 3$, $$(2^n-1)^{\frac{n}{2(n-1)}}\geq (2^{n-1}-1)^{\frac{n-1}{2(n-2)}}+1.$$ I've tried basic algebra and induction, but the inductive hypothesis ...
-1
votes
2answers
50 views

Trigonometric Identities dilemma

If $\cos^2 x + \sin^2(x) =1 $ Does $\cos2x = \cos^2(x) - \sin^2(x) = 1$ too? meaning $\cos^2(x) - \sin^2(x) = 1$ and $\cos^2 x + \sin^2(x) =1$ How so? It doesn't make sense to me.
1
vote
1answer
83 views

Analog to Euler's Reflection Formula (Connecting the Gamma Function to Cosines)

I'm wondering if there is an analogous equation describing the relation between the Gamma function and cosines, as Euler's Reduction Formula does for the Gamma function and sines: $\Gamma (z) \Gamma ...
2
votes
0answers
30 views

Is it always possible to go from one identity to another?

This question was inspired by this Quora question. I'm sure lots of you are familiar with the fact that we have many different representations of $\pi$, things like $$ \begin{align} \pi & = ...
3
votes
1answer
38 views

Identity for natural log of matrices

Let $A$ and $B$ be square matrices such that $\ln (A)$ , $\ln (B)$ and $\ln (AB)$ are all defined. Is it true that $$\ln (AB)= \ln (A) + \ln(B)$$ only if $AB=BA$ ? I appreciate any help, Thanks
9
votes
2answers
101 views

proof of $-\ln\left(2\sin\left(\frac x2\right)\right)=\sum_{k=1}^\infty \frac {\cos(kx)}{k}$?

There is an identity that I have seen pop up in a few questions on this stackexchange, and I was wondering what proof there is for it. It goes something like this: $$-\ln\left(2\sin\left(\frac ...
6
votes
1answer
71 views

Theoretical way to prove no positive integer $n$ exists such that $n+3$ and $n^2+3n+3$ are both perfect cubes.

I have to prove that for any positive integer $n$ at least one of $n+3$ and $n^2+3n+3$ is not a perfect cube. Is there a methodical way to solve this problem? I managed to solve it by contradiction, ...
1
vote
1answer
29 views

Permutations, compositions and associativity properties

Let n be a postive integer, and let σ : {1, . . . , n} → {1, . . . , n} be a one-to-one and onto map. Then σ is called a permutation on n elements. The set of all permutations on n elements is denoted ...
30
votes
2answers
843 views

Meaning of the identity $\det(A+B)+\text{tr}(AB) = \det(A)+\det(B) + \text{tr}(A)\text{tr}(B)$ (in dimension $2$)

Throughout, $A$ and $B$ denote $n \times n$ matrices over $\mathbb{C}$. Everyone knows that the determinant is multiplicative, and the trace is additive (actually linear). \begin{align*} \det(AB) = ...
0
votes
2answers
10 views

Properties of exponents when dealing with induction.

This will most likely be a simple question for most of you. While watching my lecture today the white board cut out and the instructor didn't explain the final step in an example. He went from ...
0
votes
1answer
18 views

Show that $B^{(1)} = 0$ and $B^{(2)}$ have basis $\{[x_i,x_j]; i>j\}$

Definition: A polynomial $f \in K \langle X \rangle$ is called a proper polynomial, if it is a linear combination of products of commutators: $$f(x_1,x_2,...x_n) = \sum ...
1
vote
1answer
43 views

Prove that $\log_a(1/x)=-\log(x)$.

I thought to write $$\log_a(1/x)=\log_a(x^{-1})=-\log_a(x)$$. But it has two problems: when x.0 and on the other problem it doesn't mention any condition. How should I solve it in each of them?
0
votes
5answers
62 views

Proving a complicated identity

Prove I know how to solve it, yet I can't! first I join fractions (Easy) then I "express" tans in sines and cosines after it everything turns black!
1
vote
2answers
42 views

Simplifying trig expression

I have $$\frac{\tan{15^\circ}}{1-\tan{15^\circ}^2}$$ and need to simplify it. The only equation I have that is even close to a match for it is $2\frac{\tan{15^\circ}}{1-\tan{15^\circ}^2}$. But the ...
5
votes
0answers
81 views

Multiplying three factorials with three binomials in polynomial identity

I have checked the following identity (1) below for $n\leq 40$ with a computer. Let $(n)_k$ denote the falling factorial $n(n-1)\ldots (n-k+1)$, let $Z_n=\sum_{k=0}^n (n)_k x^{n-k}$, and finally let ...
0
votes
0answers
16 views

Wilf-Zeilberger context with an extra parameter

Define two sequences $(A_p(X)),(B_p(X))$ of polynomials by $A_p(X)=(-2p-8)X^2+(3p^2+22p+40)X-(p^3+11p^2+40p+48)$ and $B_p(X)=(4p+12)X^2-(3p^2+21p+34)$. Let $(g_p)_{p\geq 1}$ be the sequence of ...
0
votes
2answers
50 views

Prove following equation is an identity problem

I need to 'prove' that $(1+ cot(x))^2 - 2cot(x) = 1/((1-cos(x))(1+cos(x)))$ The book doesn't actually show answers for these types of problems, which hasn't been a problem till now, I've found the ...
3
votes
4answers
73 views

Prove that $ab \leq \frac14$ and $ (1+1/a)(1+1/b)\ge 9$ when $a+b=1, a \ge 0, b \ge 0$

Our teacher gave us some identities firstly and said we could use one of them to prove it. The identities are: $$\frac{(a^2+b^2)}{2}≥(\frac{(a+b)}{2})^2$$ $$(x+y)^2≥2xy$$ and $$\frac{(x+y)}{2} \ge ...
1
vote
1answer
38 views

Deriving an identity using the Woodbury matrix identity

I am working through an algorithm derivation in Kernel Adaptive Filtering: A Comprehensive Introduction by Liu, Principe and Haykin. The part I'm having trouble with is on page 104 if you have the ...
2
votes
10answers
153 views

How to prove $x^3-y^3 = (x-y)(x^2+xy+y^2)$ without expand the right side?

I can prove that $x^3-y^3 = (x-y)(x^2+xy+y^2)$ by expanding the right side. $x^3-y^3 = (x-y)x^2 + (x-y)(xy) + (x-y)y^2$ $\implies x^3 - x^2y + x^2y -xy^2 + xy^2 - y^3$ $\implies x^3 - y^3$ I was ...
3
votes
2answers
46 views

Prove that $ \cos x - \cos y = -2 \sin ( \frac{x-y}{2} ) \sin ( \frac{x+y}{2} ) $

Prove that $ \cos x - \cos y = -2 \sin \left( \frac{x-y}{2} \right) \sin \left( \frac{x+y}{2} \right) $ without knowing cos identity We don't know that $ \cos0 = 1 $ We don't know that $ \cos^2 x + ...
1
vote
2answers
58 views

Trig Identity / Pythagorean Theorem confusion?

I run into a problem when I'm trying to prove how $\tan^2x+1 = \sec^2x$, and $1+\cot^2x=\csc^2x$ I understand that $\sin^2x+\cos^2x = 1$. (To my understanding 1 is the Hypotenuse, please correct me ...
4
votes
2answers
89 views

Why not add something to both sides of a purported identity to prove it? [duplicate]

A section in my precalculus book is devoted to establishing (=proving) trigonometric identities, and a typical problem in the book presents a purported identity and asks students to establish it. The ...
1
vote
2answers
44 views

Proving $\sum_{k=0}^{n} {n \choose k} = 2^n$ with Newton's Binomial Theorem

I'm having a hard time proving this theorem from a textbook. Theorem For any integer $n \ge 0$, we have $$\sum_{k=0}^{n} {n \choose k} = 2^n$$ Proof Take x = y = 1 in Newton's Binomial Theorem My ...
8
votes
3answers
394 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
49 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
104 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
41 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
62 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
35 views

Simplifying a complex exponential equation

Can Someone please explain which identities are required to show that Thank you
6
votes
5answers
192 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
72 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
41 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
32 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
33 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
231 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
38 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
19 views

Iterated sums identity

How to show that the following iterated sums are equal? $\sum_i \sum_j f(i)h(j)g(i,j) = \sum_j\bigg(\sum_i g(i,j)f(i)\bigg) h(j)$
3
votes
7answers
167 views

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

Are there two functions $f$ and $g$ s.t. $$f \circ g = \operatorname{id}$$ but $$g \circ f \neq \operatorname{id}?$$ Could someone give an example or a proof that this is impossible? This must be ...
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 ...
0
votes
1answer
41 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
49 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
21 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