Identity Definition and 1000 Threads

I'm trying to prove this problem out of Allan Clark's Elements of abstract algebra. Given an epimorphism \phi from R -> R' Prove that: \phi^{-1}(a'b') = (\phi^{-1}a')(\phi^{-1}b') where a' and b' are ideals of R' I had no trouble showing that (\phi^{-1}a')(\phi^{-1}b') is a subset...

Let T_{n} = n*(n+1)/2 and n and m are integers. I discovered that 2*n+1 = \frac{(T_{(n-1)} -m)*(T_{(n+2)}-m) - (T_{(n-2)}-m)*(T_{(n+1)}-m)}{(T_{n} - m - 1)} except for the case where the denominator is zero. Is there a simple way to prove this identity?

Homework Statement Is the identity C(m, a) + C(m,a+1) = C(m+1,a+1) (where C is the binomial coefficient function) a special case of Vandermonde's identity: \sum_{k=0}^r \binom{m}{r-k} * \binom{n}{k} = \binom{m+n}{r} Homework Equations The Attempt at a Solution n (or m) must...

Homework Statement Evaluate the integral: (int sign) Sech³xTanhx dxHomework Equations Derivative of Sechx = -(SechxTanhx)The Attempt at a Solution Rewrite as: Sech²xSechxTanhx U=sechx Du = -(SechxTanhx)dx -Du = SechxTanhx dx replace into integral -(integral sign) U²du Evaluate: -U³ / 3...

Hey guys, Could someone give me a clear explanation of what Vandermonde's Identity is? I'm looking at the proof in my book and I'm having a difficult time understanding this. Fortunately I understand the rest of the section (which covers Binomial theorem, Pascal's identity and triangle)...

Hi. I was just wondering, how can i prove the following identity: \frac{{dy}}{{dx}}\frac{{dx}}{{dy}} = 1 Its nothing that I am required to know, but i was just curious, so for all i know, it may be way out of anything that i can mathematically comprehend. The best I've been able to...

The textbook states something along the lines as prove the identity. 1 - ((sin^2x)/(1+cos x)) = cos x If you want you an work this out algebraically relatively easily to get cos x = cos x. But what if you put pi back into the original equation? You get 1 - undefined = cos x. So I graphed...

I got some precalc review to prepare for calc, and after hours of doing the packet, I'm on the last problem set...but it's all about derivatives which we never touched on last year. Homework Statement I'm supposed to derive sin^2 + cos^2 = 1 Homework Equations It says to use cos 0...

Homework Statement To prove that (∂u/∂T)_P=c_P – Pβv where _P =>P constant;β=>co-eff. of vol exp. Homework Equations The Attempt at a Solution I proved it for ideal gases. Write d'Q=dh-vdP Now expand d'Q with 1st law and du(in 1st law) in terms of dP and dT.Since du is a total...

For a homework assignment I'm supposed to prove that sin(x)^2+cos(x)^2=1, using only the following identities (along with algebraic operations): sin(-x)=-sin(x) cos(-x)=cos(x) cos(x+y)=cos(x)cos(y)-sin(x)sin(y) sin(x+y)=sin(x)cos(y)+cos(x)sin(y) I can't figure this out, because as far...

From http://www.us.oup.com/us/catalog/general/subject/Physics/Relativity/?view=usa&ci=9780198567325" I learn that the fundamental identity c^2\text{d}{t'}^2 -\text{d}{x'}^2 -\text{d}{y'}^2 -\text{d}{z'}^2 = c^2\text{d}{t}^2 -\text{d}{x}^2 -\text{d}{y}^2 -\text{d}{z}^2 relates co-ordinates...

How to prove g^{im} \nabla_{\partial_j}R_{ilkm}=\nabla_{\partial_j}R_{lk}. of cause g^{im}R_{ilkm}=R_{lk}, but I don't know how the contraction can pass through the covariant derivative?

Homework Statement we are to show a=(1/2) closed loop integral over [r x dl] Homework Equations The Attempt at a Solution I suppose this can be done formally from the alternative form of Stokes' theorem that can be obtained by replacing the vector field in curl theorem by VxC...

The question is "Verify that each equation is an identity." Problem tan 8k - tan 8k tan^2 4k = 2 tan 4k The first thing I tried was to factor out the tangents. tan 8k (1 - tan^2 4k) then if I'm doing this correctly tan 8k (1 - tan 2k) (1 + tan 2k) But from here I'm stuck, that is if I'm on...

Which trigonometric identitiy are used to transform 2(cos(x))^2+2cos(x)sin(x) into 0?

Homework Statement http://www.jyu.fi/kastdk/olympiads/2004/Theoretical%20Question%203.pdf http://www.jyu.fi/kastdk/olympiads/2004/Solution%203.pdf Question A- (b) They use some trigomentric identity that I don't understand, which one is it? Thanks in advance. Homework...

Hypothetically- If you were able to split in two like a bacterium, would you be two people with separate identities, or would you be one person living in two bodies?

Can someone point me in the right direction with vandermonde's identity, I'm seeking a algebraic proof. essentially it its the sum of (a ,k)(b ,n-k) = (a+b ,n) when summed over k = 0 to n. Could someone right this out in latex since it is probably incomprehensible. I used (a ,k) to denote...

Homework Statement Is this correct (in the document)? The Attempt at a Solution I have a feeling it is not.

Question Statement If A+B+C = 180, prove that cos (A+B-C) + cos (B+C-A) + cos (C+A-B) = 1+4 cosAcosBcosC My Attempt If A+B+C=180, Then A+B-C=180-2C cos (A+B-c)=cos(180-2C) (After some substitution and caculation) cos (A+B-C) = -cos 2C Similarily, I obtain the same expression for...

Using the theorem that in any boolean ring a+a=0 for all a in boolean ring R. Then 0 is in R. Make the multiplicative identity 1 is also in it. Therefore R can only take 0 and 1 and no more because 1+1=0. 0+0=0. 1+0=1 always. So 2 or other elements can never occur.

Homework Statement hi, I'm working on constructible things again and in one of the proofs our prof threw out this identity and I just don't know where it came from...

Homework Statement If dU=TdS-PdV then (dS/dV)T=(dP/dT)V the T and V at the end means that T and V are constant How did they get this identity? It came from a thermodynamics hence for their notations. I have tried ways like rearranging but it dosen't seem to work. I think it has something to...

Homework Statement I am to show: closed integral {phi (grad phi)} X (n^)dS=0 Homework Equations The Attempt at a Solution I understand I am to use divergence theorem here.but cannot approach.Please help

Homework Statement Prove the Identity: tan²θ - sin²θ = tan²θsin²θ Homework Equations The Attempt at a Solution Well I got up to (sin²θ/cos²θ) - (sin²θcos²θ/cos²θ) = sin⁴θ/cos²θ I got the answer cause my calculator says sin²θ - sin²θcos²θ = sin⁴θ But I don't know how sin²θ - sin²θcos²θ =...

just wondering... does the parity operator squared give the identity operator?

Homework Statement Prove the following identity: (1 + cos \theta)^2 + sin^2\theta = 2(1 + cos \theta) Homework Equations cos^2 \theta + sin^2 \theta = 1 The Attempt at a Solution I've squared out the first bracket so that it becomes 1 + cos^2 \theta and multiplied out the second bracket so...

Can someone guide me on how to prove that F_{4n+3} + F_{4n+6} = F_{2n+1}^2 + F_{2n+4}^2 either side of the above is the difference (F_{2n+2}*F_{2n+3} + F_{2n+4}^2) - (F_{2n}*F_{2n+1} + F_{2n+2}^2) I intend to post this sequence F_{2n}*F_{2n+1} + F_{2n+2}^2, with a comment re a few...

I'm trying to solve this trig problem: sin^2000(x) + cos^2000(x) = 1 I'm not sure how to go about it... I tried starting with sin^2(x) + cos^2(x) = 1 and build up to 2000 but I didn't get very far. Obviously any multiple of pi will be an answer since either sin^2000 or cos^2000 will be...

Hi. I need to prove the following identity \arccos{z} =i \ln { z + (z^2 -1)^\frac{1}{2} } I was given a hint to write \cos{A}=z, then rewrite \cos{A} in terms of the exponential. \cos{A}=\frac{\exp{iA}+\exp{-iA}}{2}=z I took the log on both sides and got stuck at that...

I am having problems with one exam question. Does this diophantine equation have a solution(s) 12a+21b+33c=6 as far as I know this is not a linear equation, and what I read online says that Bezout identity only applies for linear diophantine equations. The solution says gcd(12,21,33)...

Homework Statement On the set of Natural Numbers from 1 to 10000 are given the following identity relations. R1 ; n R1 m where m and n have the same remainder by division by 24, that is mod n 24 == mod m 24. R2 ; n R2 m where n and m have in decimal notation the same number of 2s R3; n...

In my book, (cos4x)^2 is written 1+cos8x without referring to any formula. Which trig. identity is used here?

Can anyone help me solve the following problems? sec theta -1/1-cos theta = sec theta tan (pie/2 - theta) tan theta =1 Thanks

Hi can someone help me prove this identity? I'm having trouble understand the role of the interior product or more precisely how to calculate with it. My professor uses the "cut" notation _| but i don't see this in any textbooks. Can someone give me some hints on how to prove the identity?

(1-xt)/(1-2xt+t^2)=sum(Tn(x)t^n) How can i prove this equation? Could you give me a hint or suggestion?

Homework Statement Use the thermodynamic identity to derive the heat capacity formula C_V=T\frac{\partial{S}}{\partial{T}}_VHomework Equations C_V=\frac{\partial{U}}{\partial{T}} T=\frac{\partial{U}}{\partial{S}} dU=TdS-PdV+\mudN The Attempt at a Solution I used...

Homework Statement Show that, for any two nonzero complex numbers z_1 and z_2, \text{Log } (z_1 z_2) = \text{Log } z_1 + \text{Log } z_2 + 2 N \pi i \, , where N has one of the values -1, 0, 1. Homework Equations The logarithm on the principal branch is: \begin{align*}...

There is one last problem i have on my trig assignment and i have no clue how to do it. The questions is: Find sinx, cosx, tanx, sin2x, cos2x, and tan2x from the given information: secx=5, sin is negative. If anyone could show me how to do this problem it would be soooo appreciated...

Hey I've got an assignment on trig identities and can't figure this one out. Prove the Identity: tanx+cotx=2csc2x I got to tanx+cotx= 1/2 sin2x=1/4sinx2cosx but when i get to the point where i have numbers in front of the sinx or cosx i don't know what to do. Thanks for any help

Today the professor wrote down the following for use in an integral: \frac{1}{2}cos(2\theta)=sin\theta cos\theta i am not sure is this is correct. i have tried to prove this and cannot, i have also not found it in an table of trig identities. Is this a valid identity? i am not sure since...

Question: If \hat{A} and \hat{B} are two operators such that \left[\hat{A},\hat{B}\right] = \lambda, where \lambda is a complex number, and if \mu is a second complex number, prove that: e^{\mu\left(\hat{A}+\hat{B}\right)}=e^{\mu\hat{A}}e^{\mu\hat{B}}e^{-\mu^{2}\frac{\lambda}{2}} What I...

Is there a non-ugly proof of the following identity: \langle Ax,y \rangle = \langle x,A^*y \rangle where A is an nxn matrix over, say, \mathbb{C}, A* is its conjugate transpose, and \langle \cdot , \cdot \rangle is the standard inner product on \mathbb{C} ^n.

This isn't a homework question, it's just for personal enrichment. I've been trying to prove that \tan\frac{\theta}{2}=\frac{\sin\theta}{1+\cos\theta} I tried starting off with \tan\frac{\theta}{2}=\frac{\sin\frac{\theta}{2}}{\cos\frac{\theta}{2}} is this even the right way to start the...

tan3A= 3tanA-tan3A/1-3tan^2 A

could anyone explain (no calculation) how to work the following identity so could get an understanding of it thank you tan(45+A)degrees tan(45-A)degrees=1

can anyone help[ me prove the following identity i keep on ending up in a dead end tan3A=3tanA-tan^3 A/ 1-3tan^2 A thank you

Hey all, I am unsure how to do this problem... i find problems where i have to derive things quite difficult! :P http://img143.imageshack.us/img143/744/picture2ao8.png this is the Full Fourier series i think and so the Fourier coeffiecients would be given by...

Hi, I am trying to prove \frac{tanx+1}{tanx-1}=\frac{secx+cscx}{secx-cscx} I am working on the right side and trying to get it to the same form as the left side. I get to \frac{1+2sinxcosx}{sin^{2}x-cos^{2}x} which I COULD apply the fact that sin2x=2sinxcosx as well as...

Hi, how do I interpret the last sum: http://planetmath.org/encyclopedia/LagrangesIdentity.html Sum (...) 1<=k < j <= n Is it the double sum: Sum( Sum( (a_k*b_j - a_j*b_k)^2 from k = 1 to n) from j = 2 to n ) ?