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. Francisco Alegria

    A Identity with Gamma matrices and four-vector contractions

    Is the fowwowin identity correct for a generic four-vector"q"? What is the proof? Thank you.
  2. Theia

    Geometry Looking for a good book about trigonometry

    Hi all! I've never been studied the identities and such of secant, cosecant and cotangent. Yet I think, it would be useful to have them in my toolbox. Thus I'm asking, if anyone would know a reasonable book or other kind of material (paper or pdf) about trigonometry that has brief theory...
  3. C

    Another way to find trig identities

    Using the identity's (1) (2) Gives, Why dose this elegant method work? Many thanks!
  4. C

    Proving trig identities -- Is the method related to the unit circle?

    Why when proving trig identities, Do we assume that r = 1 from ## rcis\theta = r[\cos\theta + i\sin\theta]##? This makes me think that this is somehow it is related the unit circle. Note: I am trying to prove the ##cos3\theta## identity and am curious why we assume that the modulus is 1...
  5. chwala

    Proof of the trig identities for half-angles

    I was just checking this out the sin##\frac {A}{2}## property, in doing so i picked a Right-Angled triangle, say ##ABC##, with ##AB=5cm##, ##BC=4cm## and ##CA= 3cm##. From this i have, ##s=6cm## now substituting this into the formula, ##sin\frac {A}{2}##= ##\frac {1×3}{5×3}##=##\frac...
  6. D

    MHB Can you prove the following two difficult trigonometric identities?

    Can you prove the following? [sec(x)]^6 - [tan(x)]^6 = 1 + 3*[tan(x)]^2*[sec(x)]^2 [sin(x)]^2*tan(x) + [cos(x)]^2*cot(x) + 2*sin(x)*cos(x) = tan(x) + cot(x) If not, the following free math tutoring video shows you the method:
  7. B

    MHB Problems involving Trigonometric Identities

    What are the step-by-step in solving these problems?
  8. chwala

    Proving trigonometry identities

    I was just looking at the problem below: there may be several ways to prove the identity: question: Mark scheme solution: My take: we may also use ##sin^{2}x+cos^{2}x≡(sin x+ cos x)(sin x-cosx)##... we end up with(##\frac 2 {\sqrt{2}}##cos ∅)(##\frac 2...
  9. V

    4cos(2x) = 8sin(x)cos(x) -- Help with identities

    4cos2x = 8sinxcosx 4cos2x - 8sinxcosx = 0 Now I am stuck. I don't know what identities to use. I can see it was set to 0 for a reason. But why? I know the answer is 4 - 4tan2x = 0 but how? Thanks.
  10. F

    How to approach vector calculus identities?

    Majoring in electrical engineering imply studying Griffiths book on electrodynamics, so I have begun reading its first chapter, which is a review of vector calculus. A list of vector calculus identities is given, and I would like to derive each one, with one of them being ##\nabla \cdot (A...
  11. U

    MHB Solve Situational Problems Involving Trigonometric Identities

    Hi! I am so confused about the given and what is being asked, I don't know how to solve it. This topic is solving situational problems involving trigonometric identities. Your help would be a big one for me :) Thank you so much in advance!
  12. srfriggen

    B Verifying trig identities.... what about when tan is undefined?

    Hello, If I wanted to verify tan(x)cos(x) = sin(x), what about when x is pi/2? LHS has a restricted domain so it can't equal sin(x). Does this equation only work with a restricted domain? Nothing in my textbook discusses that. Thank you
  13. JD_PM

    Showing properties of a propagator given certain Lorentz identities

    The following exercise was proposed by samalkhaiat here. The given Lorentz identities were proven here. We first note that ##d^4 k = d^3 \vec k dk_0##, the ##k_0## integration is over ##-\infty < k_0 < \infty## and ##\epsilon (k_0)## is the sign function, which is defined as $$\epsilon...
  14. JD_PM

    I Proving Lorentz Transformation identities

    This exercise was proposed by samalkhaiat here Given the defining property of Lorentz transformation \eta_{\mu\nu}\Lambda^{\mu}{}_{\rho}\Lambda^{\nu}{}_{\sigma} = \eta_{\rho \sigma}, prove the following identities (i) \ (\Lambda k) \cdot (\Lambda x) = k \cdot x (ii) \ p \cdot...
  15. MichaelRocke

    Trig Identities - Pre-calculus in a Nutshell - Section 4 Question 1

    My latest attempt \begin{align*} \frac{\sin \theta + \tan \theta}{\csc \theta + \cot \theta} = \\ \frac{\sin \theta + \tan \theta}{\csc \theta + \cot \theta} \cdot \frac{\csc \theta - \cot \theta}{\csc \theta - \cot \theta} =\\ \frac{\sin \theta \csc \theta + \tan\theta \csc \theta - \sin...
  16. benorin

    I Hypergeometric Functions Identities: n_F_n & (n+1)_F_n

    See attachment for identities and proofs, if you find my proofs are incorrect in some way please post it. Thanks for your time.
  17. M

    B Why do we care about trig identities?

    Homework Statement: This is not a homework question. I am trying to understand why we spend so much time studying trig identities. Homework Equations: As far as I understand, the two basic trig functions (sin and cos ) show the relationship between the sides of a right angle triangle in a...
  18. CharlieCW

    How do I deduce some basic thermodynamic identities using multivariate calculus?

    Homework Statement Let x, y and z satisfy the state function ##f(x, y, z) = 0## and let ##w## be a function of only two of these variables. Show the following identities: $$\left(\frac{\partial x}{\partial y}\right )_w \left(\frac{\partial y}{\partial z}\right )_w =\left(\frac{\partial...
  19. N

    Trigonometry - Double Angle Identities

    Homework Statement If I have the following relation: tan(2x) = (B/2) / (A - C) but tan(2x) = sin(2x) / cos(2x) How do I obtain an expression for sin(x) and cos(x) in terms of the constants, B,A,C only? Homework Equations cos(2x) = 1- 2 sin^2(x) The Attempt at a Solution [/B] I can't...
  20. chwala

    Proving Trigonometric Identities: (sin φ+1-cos φ)/(sin φ+cos φ-1)

    Homework Statement Show that ## {(tan φ+sec φ-1)/(tan φ-sec φ+1)}≡ {(1+sin φ)/cos φ}##[/B]Homework EquationsThe Attempt at a Solution ## (sin φ+1-cos φ)/(sin φ+cos φ-1)##[/B]
  21. S

    MHB How to Prove a Trigonometric Identity Involving x, y, and z?

    If xy+yz+zx=1 then prove (x^2-1)(y^2-1)/xy+(y^2-1)(z^2-1)/yz+(z^2-1)(x^2-1)/zx=4 with trigonometric identities such as cotAcotB+cotBcotC+cotCcotA=1
  22. S

    MHB Proving Identities Using Axioms of Equality

    Given the following axioms: For all A,B,C...we have: 1) A=A 2) A=B <=> B=A 3) A=B & B=C => A=C 4) A=B => A+C= B+C 5) A=B=> AC =BC ( NOTE :Instead of writing A.C or B.C e.t.c we write AB.BC e.t.c) 6) A+B= B+A..........AB=BA 7) A+(B+C) = (A+B)+C............A(BC)=(AB)C 10)...
  23. B

    I Superposition of particle identities

    Suppose a neutral meson decays into an electron and a positron. Are the two particles entangled as they fly apart? Before any measurement takes place, are the particles in a mixed superposition as to which one is the electron, and which one is the positron? Is there a way to test for such...
  24. opus

    Trig Identities -- example problem confusion

    Homework Statement Back with more trig identities. Verify that the following is an identity ##-tan\frac{a}{2} = cot\left(a\right)-csc\left(a\right)## Homework Equations All pythagorean identities, double angle formulas, half angle formulas The Attempt at a Solution In the picture that I've...
  25. J

    How Do You Simplify Trigonometric Expressions Using Basic Identities?

    Homework Statement Express (1+cot^2 x) / (cot^2 x) in terms of sinx and/or cosx Homework Equations cot(x) = 1/tan(x) sin^2(x) + cos^2(x) = 1 The Attempt at a Solution I do not know if I am solving this problem correctly. Is there an easier route than the way I have solved it, if it is solved...
  26. D

    How to prove vector identities WITHOUT using levi civita ?

    Mentor note: Thread moved from homework sections as being a better fit in the math technical section. Multiplying components of both sides are also off limits. I am trying to derive vector identities on introduction chapters in various EMT books. For example : (AXB).(CXD) = (A.C)(B.D) -...
  27. H

    B Eliminating Variables in Trigonometric Equations for Research Purposes

    Consider the following set of equations: ##r = \cosh\rho \cos\tau + \sinh\rho \cos\varphi## ##rt = \cosh\rho \sin\tau## ##rl\phi = \sinh\rho \sin\varphi## Is there some way to combine the equations to get rid of ##\varphi## and ##\tau## and express ##\rho## in terms of ##r, t, \phi##? I...
  28. Gio83

    A Derive the Bianchi identities from a variational principle?

    Einstein's field equations (EFEs) describe the pointwise relation between the geometry of the spacetime and possible sources described by an energy-momentum tensor ##T^{ab}##. As well known, such equations can be derived from a variational principle applied to the following action: $$S=\int\...
  29. binbagsss

    How Do You Solve Gamma Function Identities?

    Homework Statement To show: Homework Equations The Attempt at a Solution To be honest, I'm pretty stuck. I could try to use the third identity: ##\Gamma(-k+\frac{1}{2})=\frac{2\sqrt{\pi}}{2^{-2k}}\frac{\Gamma(-2k)}{\Gamma(-k)} ## but this doesn't really seem to get me anywhere. I...
  30. S

    Courses Trig Identities: How to Ace Calc II Without Memorizing Every Identity

    I was wondering exactly what parts of trig I need to do to do well in Calc II. I took trig this past spring and aced it and I'm taking Calc I this semester. I'm not worried about this semester because I know my instructor won't use trig outside teaching us how to take the derivatives of the trig...
  31. H

    A Problems with identities involving Legendre polynomials

    I am studying the linear oscillation of the spherical droplet of water with azimuthal symmetry. I have written the surface of the droplet as F=r-R-f(t,\theta)\equiv 0. I have boiled the problem down to a Laplace equation for the perturbed pressure, p_{1}(t,r,\theta). I have also reasoned that...
  32. G

    A Algebraic Proofs of Levi-Civita Symbol Identities

    Hello everyone, my question concerns the following: Though widely used, there does not seem to be any standard reference where the common symmetrization and anti-symmetrization identities are rigorously proven in the general setting of ##n##-dimensional pseudo-Euclidean spaces. At least I have...
  33. L

    Ward-Takahashi identities at tree level in scalar QED

    Homework Statement Let \Gamma^\mu be the three-point vertex in scalar QED and \Gamma^{\mu\nu} be the four-point vertex. Use Feynman's rule at tree level and verify that the Ward-Takahashi identities are satisfied: q^\mu \Gamma_\mu(p_1,p_2)=e[D_F^{-1}(p_1)-D_F^{-1}(p_2)],\\...
  34. M

    MHB Prove Trigonometric Identities

    I got this problem on my term test and it's the first problem I couldn't solve on a test ever since I'm in High School. I've tried to solve it at home even, but I still couldn't manage. The thing is that it doesn't even look difficult, maybe there's something I'm not seeing, so I hope someone...
  35. F

    MHB How Do You Solve These Trigonometric Identities?

    How do you do this one? I can't figure it out! (2 - 5cot x) / (2 + 5cos x) = (2sin x - 5cos x) / (2sin x + 5cos x)
  36. Faisal Moshiur

    I Proof of some identities regarding spin angular momentum.

    If we define Si=(1/2)× (reduced Planck's const)×sigma Then what will be (sigma dot vect{A})multiplied by (Sigma dot vect{B}) Here (sigma)i is Pauli matrix. Next one is, what will we get from simplifying <Alpha|vect{S}|Alpha> where vect{S} is spin vector & |Apha>is equal to " exp[{i×(vect{S} dot...
  37. G

    Identities of fields in Maxwell's equations

    Hello. I would like to ask one simple question. Do we need to distinguish E-field (Electric field) in Gauss's law from those in Maxwell-Faraday equation and Ampere's circuit law? I firstly thought that E-field in Gauss's law is only for electrostatics so I need to distinguish it from E-field in...
  38. B

    A How Do Gamma Matrix Identities Relate to the Charge Conjugation Operator?

    Consider the matrix ##C = \gamma^{0}\gamma^{2}##. It is easy to prove the relations $$C^{2}=1$$ $$C\gamma^{\mu}C = -(\gamma^{\mu})^{T}$$ in the chiral basis of the gamma matrices.1. Do the two identities hold in any arbitrary basis of the gamma matrices? 2. How is ##C## related to the charge...
  39. Schaus

    Proving Reciprocal Identities: (secx+1)/(sin2x) = (tanx)/2cosx-2cos2x

    Homework Statement (secx+1)/(sin2x) = (tanx)/2cosx-2cos2x) Homework EquationsThe Attempt at a Solution Left Side ((1+cosx)/cosx)/2sinxcosx ((1+cosx)/cosx) x (1/2sinxcosx) cancel the a cosx from both to get (1/2sinxcosx) This is all I could manage with left side so I tried right side Right...
  40. S

    MHB How can the quadratic equation be used to solve a trigonometric identity?

    Can anybody please help me solve this? 4cot² - 6 cosec x = -6
  41. S

    MHB How can you solve equations involving trigonometric identities?

    Can anybody please help me solve either of these equations Solve the following equation for angles between 0 and 360 degrees 4cos²θ + 5sinθ = 3 4cot² - 6 cosec x = -6
  42. S

    B Factoring quadratic equation (with trig identities used)

    Is it possible to factor a quadratic equation along the lines of asin^2x -bsin2x+c ? If so, how? The sin2x seems to be a problem since when expanded it becomes 2sinxcosx, but I'm wondering if it is possible, and how it would be done? Thanks in advance.
  43. A

    Trig identities (I think?) for precalc review

    Homework Statement My calc class is having me review precalc(which I'm really rusty on...) 21. Find sin θ, sec θ, and cot θ if tan θ = 27 22. Find sin θ, cos θ, and sec θ if cot θ = 4. 23. Find cos 2θ if sin θ = 15 24. Find sin 2θ and cos 2θ if tan θ = √2 25. Find cos θ and tan θ if sin θ =...
  44. Evangeline101

    Is My Trigonometric Identity Proof Correct?

    Homework Statement Homework Equations none The Attempt at a Solution [/B] I literally just posted this in the thread: https://www.physicsforums.com/threads/proving-identities.881951/ But since it was marked solved I doubt anyone will see it. So sorry in advance for making a new thread on...
  45. alexandria

    How Do You Prove Trigonometric Identities with Minimal Equations?

    Homework Statement prove the following identity [/B] Homework Equations no equations required The Attempt at a Solution I've been trying to prove this identity, but no matter what I do, I can't seem to make both sides the same here is my answer to this qts so far: can someone please tell me...
  46. awholenumber

    I Question ,trigonometric identities equation and functions ?

    what is the difference between trigonometric identities , equations and functions ...? is it possible to apply some numerical method on a trigonometric function ?? i was looking for an example where numerical methods could be applied on a trigonometric function ... i am not sure what you...
  47. L

    A Difficulty in understanding contracted Bianchi identities

    I am confused about the contraction in the proof of the contracted Bianchi identities in https://en.wikipedia.org/wiki/Proofs_involving_covariant_derivatives from the step {g^{bn}}(R_{bmn;l}^m - R_{bml;n}^m + R_{bnl;m}^m) = 0 it seems that the following two quantities are equal...
  48. V

    Why Does This Trigonometric Identity Seem Incorrect?

    Homework Statement sin^2x + 4sinx +4 / sinx + 2 = sinx +2 Homework EquationsThe Attempt at a Solution L.S = sin^2x + 4sinx +4 / sinx + 2 =1-cos^2+4(sinx + 1) / sinx +2 Not sure where to go from there. Not sure if I was even supposed to factor out the 4?
  49. R

    Vector identities in quantum mechanics

    The overall problem is to prove that [L^2,[L^2,\hat{r}]]=2\hbar^2 {L^2,r} I feel I am very close to solving this problem but I need a quantum version of the vector identity ax(bxc). Because the relevant vectors are operators that don't commute, there is a problem. Does anybody know of a source...