Identity Definition and 1000 Threads

Homework Statement Given a triangle ABC prove that c sin \frac{A-B}{2} = (a-b)cos \frac{C}{2} Homework EquationsThe Attempt at a Solution It looks rather similar to a formula mentioned in my book's lead-in to this exercise: \frac{a-b}{a+b}=tan\frac{A-B}{2}tan\frac{C}{2} Which can be...

Hi, I need help proving the following trig identity, (2sinx)\overline{secxtan(2x)}=2cos^2x-csc^2x+cot^2x I have tried starting from the left hand side, the right hand side, and doing both together, but nothing seems to work. One of the ways I tried: LHS...

Hi, I need help proving the following trig identity: $$\frac{\cot^2(x)-\cot(x)+1}{1-2\tan(x)+\tan^2(x)}=\frac{1+\cot^2(x)}{1+\tan^2(x)}$$ Me and my friend have spent several hours determined to figure this out, starting from the left hand side, the right hand side, and doing both together...

How prove $\cos\frac{8\pi}{35}+\cos\frac{12\pi}{35}+\cos\frac{18\pi}{35}=\frac{1}{2}\cdot\left(\cos\frac{\pi}{5}+\sqrt7\cdot\sin\frac{\pi}{5}\right)$?

Homework Statement For a single molecule, derive the internal energy U = 3/2kBT In terms of the partition function Z, F = -kBTlnZ Where Z = V(aT)3/2 Homework Equations Thermodynamic identity: δF = -SδT - pδV p = kBT/V S = kB[ln(Z) + 3/2]The Attempt at a Solution U = F + TS δU = δF +...

I've worked with Euler's Identity for physics applications quite a few times, but ran into a "proof" of a contradiction in it, which I can't seem to find a flaw in (the only time I've ever had to do any proofs was in high school). I've derived Euler's equation in two different ways in past...

Prove the following identity: \frac{1-tan^2(\theta)}{1-cot^2(\theta)}=1-sec^2(\theta) By tan/sec identity, \frac {2-sec^2(\theta)}{2-csc^2(\theta)} separated the variables \frac {2}{2-csc^2}- \frac {sec^2}{2-csc^2} There's absolutely nothing useful I can do with either of those terms, so...

Mod note: Fixed the LaTeX. ##a=sinθ+sinϕ## ##b=tanθ+tanϕ## ##c=secθ+secϕ##Show that, ##8bc=a[4b^2 + (b^2-c^2)^2]## I tried to solve this for hours and have gotten no-where. Here's what I've got so far : ##a= 2\sin(\frac{\theta+\phi}{2})\cos(\frac{\theta-\phi}{2}) ## ## b =...

Homework Statement For the space Lw2(a,b), we can write the basis in a discrete fashion as {en|n∈ℤ} or in a continuous fashion as |x> (as we would in quantum mechanics for the position representation), such that we may write the identity operator as either I=∑n|en><en| or I=∫abdxw(x)|x><x|...

Prove the following $$\sum_{k=1}^n \frac{H_k}{k} = \frac{H_n^2+H^{(2)}_n}{2}$$ where we define $$H^{(k)}_n = \sum_{j=1}^n \frac{1}{j^k} \,\,\, ; \,\,\, H^2_n = \left( \sum_{j=1}^n \frac{1}{j}\right)^2 $$

Hey fellahs, got another whopper that's killing me. $$1 - \cos2\theta + \cos8\theta - \cos10\theta=?$$ My objective here is to complete the identity, and my worksheet lists the correct solution as: $$4\sin\theta\cos4\theta\sin5\theta$$ And once again I've had trouble beating this one. This...

Why ##\sin^2 2x + \cos^2 2x = 1##? Will ##\sin^2 3x + \cos^2 3x## or ##\sin^2 4x + \cos^2 4x## and so on, be = 1? How to proof this?

Hi I'm taking a math course at university that covers introductory group theory. The textbook's definition of the identity element of a group defines it as two sided; that is, they say that a group ##G## must have an element ##e## such that for all ##a \in G##, ##e \cdot a = a = a \cdot e## ...

Homework Statement I am trying to prove $$\vec{\nabla}(\vec{a}.\vec{b}) = (\vec{a}.\vec{\nabla})\vec{b} + (\vec{b}.\vec{\nabla})\vec{a} + \vec{b}\times\vec{\nabla}\times\vec{a} + \vec{a}\times\vec{\nabla}\times\vec{b}.$$ I can go from RHS to LHS by writng...

Homework Statement IF G, H and G+H are invertible matrices and have the same dimensions Prove that G(G^-1 + H^-1)H(G+H)^-1 = I 3. Attempt G(G^-1 +H^-1)(G+H)H^-1 = G(G^-1G +G^-1H + H^-1G + H^-1H)H^-1 = (GG^-1GH^-1 +GG^-1HH^-1 +GH^-1GH^-1 +GH^-1HH^-1) = GH^-1+I +GH^-1GH^-1 +GH^-1 =2GH^-1+...

Prove that $$\lfloor nx \rfloor = \sum_{k=0}^{n-1}\lfloor x+k/n \rfloor$$. Note $$\lfloor x\rfloor$$ means the greatest integer less than or equal to $$x$$. I proved the cases where n=2 and n=3 by writing $$x=\lfloor x\rfloor + \{x\}$$, where $$\{x\}$$ is the fractional part of $$x$$, and then...

Homework Statement Hi, I have been trying to prove one of the corollaries of the Bezout's Identity in the general form.Unfortunately,I can't figure it out by myself.I hope someone could solve the problem. If A1,...,Ar are all factors of m and (Ai,Aj) = 1 for all i =/= j,then A1A2...Ar is a...

I was reading Schwartz's qft book. I saw the proof of ward identity taking pair annihilation as an example. he claimed he didn't assume that photon is massless in this derivation. but i have confusion with this statement. gauge invariance is a fact related to massless particles. now he has...

Homework Statement [/B] This is a seemingly simple problem. All I have to do is multiply two matrices: [ 1 0 ] [ 0 1 ] (A) and [ 2 ] [ 3 ] (B) The Attempt at a Solution [/B] Because the matrix A has the same number of columns as matrix B has rows, and because matrix A is an identity matrix...

So, I would like to prove that \gamma^{\mu_{1}...\mu_{r}}=(-)^{r(r-1)/2}\gamma^{\mu_{r}...\mu_{1}} where the matrix gamma is a totally antisymmetric matrix defined as \gamma^{\mu_{1}...\mu_{r}}=\gamma^{[\mu_{1}}\gamma^{\mu_{2}}...\gamma^{\mu_{r}]} What I have done is to prove that...

For EM wave, magnetic and electrical components are in phase, meaning when E = 0, then B = 0. Thus, I understand if it is written: f(x,t) = A(cos(kx - wt) + icos(kx - wt)) Then why plane wave is always described: f(x,t) = Aei(kx-wt) = A(cos(kx-wt) + isin(kx - wt)) Implying that Real and...

Kindly help me with this problem. I'm stuck! $\frac{\csc\theta+1}{\cot\theta}=\frac{\cot\theta}{\csc\theta-1}$ this is how far I can get to $\sec\theta+\tan\theta=\frac{1}{\sec\theta-\tan\theta}$

I can't quite work out this derivation I ran into which is essentially...Asin^2(wt) + Bcos^2(wt) = A = B. Is this correct? I know that sin^2(wt) + cos^2(wt) = 1, but I can't reason out how the factoring works here? Any help?

Dear All, Please see the attachment for the text i will be referring to. what does "I = τ1 · · · τm, where each τj is a transposition acting on {1, . . . , n}. Clearly, m != 1, thus m ≥ 2. Suppose, for the moment, that τm does not fix n". what does fixing n mean in a...

Hello everyone, I'm currently writing my dissertation based around the question of whether photo editing within the spheres of fashion and beauty media and advertising rids the subjects of their identity - I have looked at numerous books about personal identity, such as Paul Ricoeur's Oneself...

Homework Statement Let S be the subset of group G that contains identity element 1 such that left co sets aS with a in G, partition G .Probe that S is a subgroup of G. Homework Equations {hS : h belongs to G } is a partition of G. The Attempt at a Solution For h in S if I show that hS is S...

Homework Statement A and B are two unit vectors in the x-y plane. A = <cos(a), sin(a)> B = <cos(b), sin(b)> I need to derive the trig identity: sin(a-b) = sin(a) cos(b) - sin(b) cos (a) I'm told to do it using the properties of the cross product A x B Homework Equations A x B =...

Homework Statement Just like my title says, we are to prove the trig identity sin^2x+cos^2x=1 using the Euler identity. Homework Equations Euler - e^(ix) = cosx + isinx trig identity - sin^2x + cos^2x = 1 The Attempt at a Solution I tried solving the Euler for sinx and cosx...

Hi, I'm hoping to clear up a few uncertainties in my mind about proving that the identity element and inverses of elements in a group are unique. Suppose we have a group \left(G, \ast\right). From the group axioms, we know that at least one element b exists in G, such that a \ast b = b \ast...

Hi there. I was following a deduction on continuum mechanics for the invariant nature of the first two laws of thermodynamics. The thing is that this deduction works with an identity, and there is something I'm missing to get it. I have the vector product: ##\vec \omega \times grad \theta##...

Am I the only one who sees the resemblance between these two identities? Schouten: <p q> <r s> +<p r> <s q>+ <p s > <q r> =0 Jacobi: [A,[B,C]]+[C,[A,B]]+[B,[C,A]]=0 In Schouten the p occours in each term in the three terms, so we can regard it as dumby variable, and somehow get a...

Homework Statement Prove that [tan(a) + 1][cot(a+pi/4) + 1] = 2 Homework Equations [tan(a) + 1][cot(a+pi/4) + 1] = 2 The Attempt at a Solution This was very hard, I tried my best at expanding. [tan(a) + 1][cot(a+pi/4) + 1] = tan(a)cot(a+pi/4) + tan(a) + cot(a+pi/4) + 1...

Homework Statement Prove that if G is a cyclic group with more than two elements, then there always exists an isomorphism: ψ: G--> G that is not the identity mapping. Homework Equations The Attempt at a Solution So if G is a cyclic group of prime order with n>2, then by Euler's...

prove: $3=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+6\sqrt{1+7\sqrt{1+8\sqrt{1+9--}}}}}}}}$

In 'Supersymmetry in Particle Physics, An Elementary Introduction', the author Ian Aitchison used for several times the following identity: λa(ζ · ρ) + ζa(ρ · λ) + ρa(λ · ζ) = 0. I know that this identity is called Schouten's Identity, which is correct when all the variables are common...

Does anyone know how to prove this identity? I don't quote understand why the associated Legendre function is allowed to have arguments where |x|>1. h_n(kr)P_n^m(\cos\theta)=\frac{(-i)^{n+1}}{\pi}\int_{-\infty}^\infty e^{ikzt}K_m(k\rho\gamma(t))P_n^m(t)\,dt where \gamma(t)=\begin{cases}...

Let S={p,q,r} and S=(S,+,*) a ring with identity. Let p be the identity for + and q the identity for *. Use the equation r*(r+q)=r*r+r*q to deduce that r*r=q. Attempt of a solution r*r=r*(r+q)- r*q =r*r+r*q - r*q But I'm not finding a clever way to deduce what is required. Any type of help...

Double integral of (52-x^2-y^2)^.5 2<_ x <_ 4 2<_ y <_ 6 I get up to this simplicity that results in a zero! 1-cos^2(@) - sin^2(@) = 0 This identity seems to be useless. HELP PLEASE.

I am attempting to work through a paper that involves some slightly unfamiliar vector calculus, as well as many omitted steps. It begins with the potential energy due to an electric field, familiarly expressed as: U_{el} = \frac{\epsilon_r\epsilon_0}{2} \iiint_VE^2dV =...

Hey! :o We know that: $$(x,x)=0 \Rightarrow x=0$$ When we have $\displaystyle{(x,y)=0}$, do we conclude that $\displaystyle{x=0 \text{ AND } y=0}$. Or is this wrong? (Wondering)

If the group G/[G,G] is abelian then how do we show that xyx^{-1}y^{-1}=1? Thanx

Hi everybody! First post!(atleast in years and years). The stationary KdV equation given by $$ 6u(x)u_{x} - u_{xxx} = 0 $$. It has a solution given by $$ \bar{u}(x)=-2\sech^{2}(x) + \frac{2}{3} $$ This solution obeys the indentity $$ \int_{0}^{z}\left(\bar{u}(y) -...

I have no idea how to go about proving this trig identiy. I mean, I've been taught that it's a safe bet to convert everything to sines and cosines, but other than that, I've no clue. Am I even on the right path?

Homework Statement Show that: curl(r \times curlF)+(r.\nabla)curlF+2curlF=0, where r is a vector and F is a vector field. (Or letting G=curlF=\nabla \times F i.e. \nabla \times (r \times G) + (r.\nabla)G+2G=0) The Attempt at a Solution I used an identity to change it to reduce (?) it to...

So, this question says "prove each identity. State any restrictions on the variables". 5a) $$\frac{sinx}{tanx} = cosx$$ I did the first part of the question correctly (proving it), but I don't understand how you determine the restrictions on the variables. In the textbook, it says that it...

hey, question is attached thanks in advance!

In a Hilbert-space whose dimensionality is either finite or countably infinite, we have the discrete resolution of identity \sum_n |n\rangle \langle n| = 1 In many cases, for example to obtain the wavefunctions of the discrete states, one employs the continuous form of the resolution...

Homework Statement From Mary Boas' "Mathematical Methods in the Physical Sciences" 3rd ed. Ch 3 Sec 4 problem 24 Where A and B are vectors. What is the value of (AXB)^2+(A dot B)^2=? Comment: This is a special case of Lagrange's Identity. Homework Equations Cross product and dot...

Hi everyone. I'm studying Heavy Quark Effective Theory and I have some problems in proving an equality. I'm am basically following Wise's book "Heavy Quark Physics" where, in section 4.1, he claims the following identity: $$ \bar Q_v\sigma^{\mu\nu}v_\mu Q_v=0 $$ Does any of you have an...

I'm slightly confused at the proof of this theorem, hopefully someone can help. Identity theorem: Suppose X and Y are Riemann surfaces, and f_1,f_2:X \to Y are holomorphic mappings which coincide on a set A \subseteq X having a limit point a \in X. Then f_1 and f_2 are identically equal. The...