What is identities: Definition and 422 Discussions

In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle.
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.

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  1. M

    Simplifying Trig Identities

    Homework Statement The length of the curve r(t) = cos^3(t)j+sin^3(t)k, 0 =< t <= pi/2 is Homework Equations AL in polar = ∫sqrt(r^2 + [dr/dθ]^2) The Attempt at a Solution I am having trouble simplifying the terms within the square root. What method should I use to deal with...
  2. A

    MHB Prove this hyperbolic identities

    6b) tanh^2(x) + 1/cosh^2(x) = 1 Could someone help start me off? I know that you have to sub in (e^x + e^-x)/2 for cosh and (e^x - e ^-x)/(e^x + e ^-x) for tanh. Then I'd add these together, but I'm not sure how I'd solve/simplify them arithmetically after that. Help would be appreciated! thanks.
  3. J

    Newton's identities and matrices

    About the Newton's identities: I'm right if I state that ek = Ik, pk = tr(Ak) and hk = det(Ak) (being Ik the kth-invariant of the matrix A)?
  4. J

    Is There Symmetry in the Signs of Newton's Identities?

    Look this relationship: http://en.wikipedia.org/wiki/Newton%27s_identities#Related_identities If I isolate the variable p, I'll have: ##p_1 = 1h_1## ##p_2 = 2h_2-h_1p_1## ##p_3 = 3h_3-h_2p_1-h_1p_2## So, my question is: BTW, would be true that: ##p_1 = 1h_1## ##p_2 = 2h_2+h_1p_1## ##p_3 =...
  5. N

    Trig identities, prove (cot(x)-tan(x)=2tan(2x))

    Homework Statement Prove cot(x) - tan(x) = 2tan(2x) Homework Equations Trig identities http://en.wikipedia.org/wiki/List_of_trigonometric_identities The Attempt at a Solution I have worked it down and don't think they are equal. I think it's supposed to be 2cot(2x) not 2tan(2x)...
  6. U

    Vector calculus identities proof using suffix notation

    I must become good at this ASAP. Homework Statement prove \vec{\nabla}\cdot (\vec{a}\times\vec{b} ) = \vec{b} \cdot(\vec\nabla\times\vec{a}) - \vec{a}\cdot(\vec\nabla\times\vec{b}) Homework Equations \vec a \times \vec b = \epsilon_{ijk}\vec a_j \vec b_k \vec\nabla\cdot =...
  7. I

    Use trig identities to show that

    Homework Statement use trig identities to show that (b) cos(tan^(−1)[x])=1/√(1+x^2) for −1/2π<x<1/2π. Homework Equations i think Pythagoras has to applied but that is geometric reasoning hmm The Attempt at a Solution
  8. K

    Proving Identities: cotx-tanx=2cot 2x

    Homework Statement Prove that cotx-tanx=2cot 2x Homework Equations Is 2cot 2x the same as 2(cos2x/sin2x) The Attempt at a Solution
  9. M

    Proving Exponential Identities

    Hi All, I am struggling to prove the following identity $$ 1 + y + \frac{1}{2!}y^2 + \frac{1}{3!}y^3 + \dots = lim_{N \to \infty} \sum_{r=0}^{N} \frac {N!}{r! (N-r)!} \frac{y}{N}^{r} = lim_{N \to \infty} (1 + \frac{y}{N})^N $$ any hint would the most appreciated. I understand the...
  10. M

    MHB Identities of the optimal approximation

    Hey! :o I am looking at the identities of the optimal approximation. At the case where the basis consists of orthogonal unit vectors,the optimal approximation $y \in \widetilde{H} \subset H$, where $H$ an euclidean space, of $x \in H$ from $\widetilde{H}$ can be written $y=(x,e_1) e_1 +...
  11. K

    Sum and differences identities equations

    Homework Statement Which is equivalent to: cos(∏/2 + x) - cos(∏/2 - x)? A) -2cos(x) B) -2 C) 0 D)-2sin(x) Homework Equations Cos (A-B) The Attempt at a Solution I am totally stuck :( please help!
  12. mesa

    Still looking for a list of Ramanujan identities, very expensive

    It is surprising how expensive the books are that contain the Ramaujan identities and equations. I understand the work to 'prove' them must have been a tremendous undertaking however I find the price of the 'notebooks' for Ramanujan's work prohibitively expensive. I believe wolfram put it...
  13. mesa

    Looking for a book with a comprehensive list of Ramanujan's identities

    I am looking for a book that has a comprehensive list of Ramaujans Identities and Equations. I have read that he created 3900 such pieces of work and it would be fascinating to see such a collection that came from just one person.
  14. S

    MHB Antiderivative involving Trig Identities.

    A little confused on something. Suppose I have the integral 2 \int 4 \sin^2x \, dx So I understand that \sin^2x = \frac{1 - \cos2x}{2} BUT we have a 4 in front of it, so shouldn't we pull the 4 out in front of the integral to get: 8 \int \frac{1 - \cos 2x}{2} \, dx then pull out the...
  15. mesa

    Need sources to search for gamma function infinite series identities.

    I have a hard time believing we only have the limited number of series I have seen so far especially considering how much broader mathematics is than I had thought just a short while ago. Where should I search to find more infinite series summations for the gamma function? For example which...
  16. mesa

    Is it okay to post just identities on arXiv?

    If I have some new identities that are very powerful but don't have time to go into the details is it okay to simply post just the functions on arXiv for now? A quick check on wolfram as to the validity of the functions will confirm all my work.
  17. A

    MHB Reduction formula instead of using identities for trigonometric integration?

    This is one of the example problems in my book to show how to deal with integrating trigonometric functions to higher powers, by breaking them down into identities. =\int cos^5x dx =\int (cos^2x)^2cos^x dx =\int (1-sin^2x)^2*d(sin x) =\int (1-u^2)^2 du =\int 1-2u^2 + u^4 du =u-\frac{2}{3}u^3...
  18. F

    Compund Angle Identities and proof

    Homework Statement prove using the compound angle identies, proove the following: \frac{sin(A-B)}{cos(A)cos(B)}+\frac{sin(B-C)}{cos(B)cos(C)}+\frac{sin(C-A)}{cos(C)cos(A)}=0 Homework Equations n/aThe Attempt at a Solution I resolved it to...
  19. S

    Proving trigonometric identities in converse

    Homework Statement If ##\sec x-\csc x=\pm p##, show that ##p^{2} \sin^2 2x +4\sin 2x-4=0## Show conversely that if ##p^{2} \sin^2 2x +4\sin 2x-4=0##, then ##\sec x-\csc x## is equal to +p and -p. Find, to the nearest minute, the two values of x in the range of 0 to 360 degrees, the equation...
  20. C

    Elegant 1/(1-x) Identities for Complex Algebraic Manipulations and Logarithms

    Found out that: 1/(1-x) = the infinite product of (1+x^(2^N)) from N=0 to infinity. From "The Harper Collins Dictionary of Mathematics" by Borowski and Borwein. Makes me wonder whether this identity could be used to make some complex algebraic manipulations into real manipulations...
  21. MarkFL

    MHB Pramod's question at Yahoo Answers regarding angle sum/difference identities

    Here is the question: I have posted a link there to this thread so the OP may view my work.
  22. ShayanJ

    Vector differential identities

    Vector differential identities! In chapter 20 of "Foundations of Electromagnetic theory" by Reitz,Milford and Christy,there is calculation which seems to make use of: \vec{\nabla}\times\dot{\vec{p}}=\Large{\frac{\vec{r}}{r}\times\frac{ \partial \dot{\vec{p}}}{\partial r}} where...
  23. M

    MHB Using the Wronskian to Solve Differential Equations with Non-Constant Solutions

    Hey! :) I need some help at the following exercise: Let v_{1}, v_{2} solutions of the differential equation y''+ay'+by=0 (where a and b real constants)so that \frac{v_{1}}{v_{2}} is not constant.If y=f(x) any solution of the differential equation ,use the identities of the Wronskian to show...
  24. skate_nerd

    MHB Proving vector calculus identities w/ tensor notation

    I have an vector calculus identity to prove and I need to use vector notation to do it. The identity is $$\vec{\nabla}(fg)=f\vec{\nabla}{g}+g\vec{\nabla}{f}$$ I tried starting with the left side by writing $\vec{\nabla}(fg)=\nabla_j(fg)$. Now I look and that and it really looks like there is...
  25. M

    Variational Principle and Vectorial Identities

    Hello there, I am struggling in proving the following. The principle of Minimum energy for an elastic body (no body forces, no applied tractions) says that the equilibrium state minimizes $$\int_{\Omega} \nabla^{(s)} u D (\nabla^{(s)}u)$$ among all vectorial functions u satisfying the...
  26. E

    Proving Some Poisson Bracket identities - a notational question

    Proving Some Poisson Bracket identities -- a notational question I need some help just understanding notation, and while this might count as elementary it has to do with Hamiltonians and Lagrangians, so I posted this here. Homework Statement Prove the following properties of Poisson's...
  27. M

    How can vector calculus 'del' relationships be derived using other identities?

    hey all! i was hoping someone could either state an article or share some knowledge with a way (if any) to derive the vector calculus "del" relationships. i.e. $$ \nabla \cdot ( \rho \vec{V}) = \rho (\nabla \cdot \vec{V}) + \vec{V} \cdot (\nabla\rho)$$ now i do understand this to be like...
  28. S

    Where can I find a list of proof identities?

    Does anyone know if there is a good place to find a list of proof identities? Basic stuff like the disjoint or if-then in logic symbols. It would be nice to have a place to make sure I'm remembering them correctly and to search for more. Thanks!
  29. N

    Trignometric functions and identities

    Homework Statement How to quickly solve problems on maximum and minimum values of trig functions with help of calculus: Ex. 10cos2x-6sinxcosx+2sin2xHomework Equations noneThe Attempt at a Solution I know the method of simplification. But i want to do it quickly with calculus. How to do that??
  30. skate_nerd

    MHB Confused about a couple of trig. identities.

    I've got this problem right now, which asks me to prove that $$Ccos(\omega_{o}t-\phi)=Asin(\omega_{o}t)+Bcos(\omega_{o}t)$$ This proved to be a bit more difficult than I expected, so I looked up a complete list of trig identities. $$cos(a\pm{b})=cos(a)cos(b)\mp{sin(a)sin(b)}$$ seems like the...
  31. TheFerruccio

    Levi-Civita Identities: Verifying Index Notation Problems

    I am having trouble establishing a process to verify various identities for problems in index notation. Description of Problem Verify that \epsilon_{ijk}\epsilon_{iqr}=\delta_{jq}\delta_{kr}-\delta_{kq}\delta_{jr} Attempt at Solution I know that the term is only positive if there is an...
  32. J

    Confusion with product-to-sum trig identities

    I'm having some confusion with a couple trig identities. On wikipedia (http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identities), the following two identities are listed: sinθcosβ = (1/2)[sin(θ+β) + sin(θ-β)] and sinβcosθ =...
  33. J

    Derivation of trigonometric identities form rotation on the plane

    Homework Statement I want to derive the trig identities starting with rotation on the plane. Homework Equations One rotation through a given angle is given by $$x' = xcosθ - ysinθ $$ $$y' = xsinθ + ycosθ$$ The Attempt at a Solution What if I wanted to rotated through any...
  34. evinda

    MHB Determining the Function with Identities

    Hi guys!I have a question..How can I show that the function that has the following identities: \bullet f(x)\neq 0 ,x\in\mathbb{R}. \bullet f(0)=f\left(\dfrac{2}{3}\right). \int_{0}^{x}\frac{f(t)f(x-t)}{f(\frac{2+t}{3})f(\frac{2+x-t}{3})}=\int_{0}^{x}\frac{f^2{(t)}}{f^2{(\frac{2+t}{3}})} is...
  35. M

    Using complex numbers to find trig identities

    I can find for example Tan(2x) by using Euler's formula for example Let the complex number Z be equal to 1 + itan(x) Then if I calculate Z2 which is equal to 1 + itan(2x) I can find the identity for tan(2x) by the following... Z2 =(Z)2 = (1 + itan(x))2 = 1 + (2i)tan(x) -tan(x)2 = 1...
  36. QuantumCurt

    Prove that the two trig identities are equivalent

    Homework Statement Prove that the two trig identities are equivalent. cos \ x \ -\frac{cos \ x}{1-tan \ x} \ = \ \frac{sin \ x \ cos \ x \ }{sin \ x \ - \ cos \ x} The Attempt at a Solution My professor recommended that we only work with one side of the equality when we're trying...
  37. T

    Diff EQ - Imaginary identities

    Homework Statement Determine the general solution of: y(6) + y''' = t The Attempt at a Solution Ok, r = 0, 0, 0, 1/2 +- 3i/√2, -1/2 + 3i/√2 What do I do with that last r value? It turns into ce-t somehow, but I don't see it. edit: typed a number in wrong, fixed now~
  38. S

    Trig identities (is my method correct?)

    Homework Statement Prove the following identities 31c) sin(\frac{\pi}{2}+x)=cosxHomework Equations sin2x+cos2x=1 The Attempt at a Solution The idea here is to prove the identity by making LS=RS so here is what i have done, but I am not sure if it is the right way, since the book shows it...
  39. B

    Can you prove the trig identities given A + B + C = 180 degrees?

    Given: A + B + C = 180 degrees Prove: TanA + TanB + TanC = TanA x TanB x TanC First time submitting something. Thank you all.
  40. E

    Summation of product identities

    Hi everybody, I am just trying to find a decent identity that relates the sum $$\sum_{k=0}^{n}a_kb_k$$ to another sum such that ##a_k## and ##b_k## aren't together in the same one. If you don't know what I mean, feel free to ask. If you have an answer, please post it. Thanks in advance!
  41. C

    How are these two equations equal? trig identities possibly?

    Homework Statement Proof that (1/6)sin(3x)-(1/18)sin(9x) = (2/9)sin^3(3x) Homework Equations The Attempt at a Solution I am just curious exactly how the power on the sine function is cubic on one side. It obviously has to do with something that increases the power on the...
  42. J

    Prove two integral identities?

    Prove two integral identities? 1. The following integral identity holds \dfrac{d}{dx}\intop_{x}^{a}\dfrac{F(\rho)d\rho}{\sqrt{\rho^{2}-x^{2}}}=-\dfrac{F(a)x}{a\sqrt{a^{2}-x^{2}}}+x\intop_{x}^{a}\dfrac{d\rho}{\sqrt{\rho^{2}-x^{2}}}\dfrac{d}{d\rho}\left[\dfrac{F(\rho)}{\rho}\right] Hints: this...
  43. E

    Product of a sequence identities

    HI, does anyone know a decent site where I can find a few product identities? I googled it, but all that came up were trig identities. I am not looking for those; I am specifically looking for product of a sequence identities: ∏
  44. M

    Proving trignometric identities.

    Homework Statement I understand this chapter a little better than the previous ones, but I'm having problems with these two problems. Can anyone at least lead me in? Homework Equations The Attempt at a Solution Starting from the right side for both. For the second one, turning the...
  45. R

    Determine Sign of Half-Angle Identities

    Homework Statement Use the figure to evaluate the function that f(x)=sin(x) f(θ/2) Homework Equations n/a The Attempt at a Solution x2+y2=1 x= -2/7 (-2/7)2+y2=1 y=√(6)/7 sin(θ/2)= +/- √(1-cos(θ)/2) (the whole function is over 2 inside of the square root) =+/-...
  46. T

    Proving tan2A = 2tanA/1-tan^2A: A Trig Identity Example

    Homework Statement Prove the following: tan2A=2tanA/1-tan^2A Homework Equations The Attempt at a Solution Took the right hand side: =2(sinA/cosA) / 1-(sin^2A/cos^2A) =2sinA/cosA /cos^2A-sin^2A/cos^2A =2sinA/cos2A /cosA/1 Dont know what to do next?
  47. MarkFL

    MHB Angle Sum/Difference Identities: Billy's Pre-calc Math Problem

    Here is the question: Here is a link to the question: Pre-calc math problem? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  48. phosgene

    Proving identities using the axioms of probability

    Homework Statement If A and B are events, use the axioms of probability to show: a) If A \subset B, then P(B \cap A^{C}) = P(B) - P(A) b) P(A \cup B) = P(A) + P(B) - P(A \cap B) Homework Equations Axiom 1: P(x)\geq 0 Axiom 2: P(S) = 1, where S is the state space. Axiom 3: If...
  49. R

    Proof the identities of the sine and cosine sum of angles

    Homework Statement I just have to prove the well known identities: \cos(\alpha + \beta)=\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta) \sin(\alpha + \beta)=\sin(\alpha)\cos(\beta)+\cos(\alpha)\sin( \beta) But the thing is that I've to use the Taylor power series for the sine and cosine...
  50. T

    Green's identities for Laplacian-squared

    Homework Statement a) Derive Green's identities in local and integral form for the partial differential operator ##\triangle^2##. b) Compute the adjoint operator ##(\triangle^2)^*##. 2. Relevant information ##U\subset\mathbb{R}^n##, ##u:U\to\mathbb{R}##, differential operator ##Lu##. In...
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