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. V

    What is the proof for 2sin2θ - 1 = sin2θ - cos2θ?

    Homework Statement 2sin2θ - 1 = sin2θ - cos2θ Homework EquationsThe Attempt at a Solution I am unsure of how to prove these. So far all I have is Left side= 2sin2θ - 1 =sin2sin2-1 And I know that right side is equal to 1. But otherwise not sure where to go from there.
  2. Ray9927

    Doubt about trigonometry Identities from sin α

    Hi all! I'm Ray and I'm new to this community, it's a pleasure! I'm trying to resolve a trigonometry exercise where I have to calculate the trigonometry Identities of a right triangle but in the specifications they don't show me any common data (hypotenuse or cathethus values), they just leave...
  3. Bill_Nye_Fan

    Help with Q7 part a: Trig Identities

    Homework Statement [/B]Q7 part a on one of the attached pictures 2. Homework Equations Trigonometric identities The Attempt at a Solution See attached pages Please help me I've spent onwards of 4 hours trying to figure this out and I can't get anywhere at all
  4. P

    Simplifying a log expression with identities

    I was supposed to simplify the expression ##\ln |\cot {x}|+\ln |\tan {x}\cos {x}|## and apparently it’s wrong. Where’s the mistake? Is it not simplified enough or . . . ? ##\ln |\cot {x}|+\ln |\tan {x}\cos {x}|## ##=\ln |\frac {\cos {x}}{\sin {x}}|+\ln |\frac {\sin {x}}{\cos {x}}\cdot \cos...
  5. T

    Proving trigonometric identities in a belt and pulley proble

    Homework Statement verify that theta in L = piD + (d-D)theta + 2Csin(theta) is equal to arc-cosine [(D-d)/2C] 2. The attempt at a solution you can see my attempt in the second picture uploaded. i don't think i even got it right
  6. C

    Trig Identities: Simplifying Expressions with Cotangent, Secant, and Cosecant

    I have re-post this forum as I should have paid closer attention to rules. I apologized for that. Homework Statement 1) The expression tan^3 θ + sinθ/cosθ is equal to: (a) cot θ (b) tan θ sec^2 θ (c) tan θ (d) sin θ tan θ (e) tan θ csc^2 θ 2) Simplify (cos θ/1+ sin θ - cosθ/sinθ-1)^-1 (a)...
  7. ognik

    MHB Playing with vector identities

    Please excuse my copying this question in, minimising my input :-) Part (b): Let $ a.b \times c =v $ Then $ a'.b' \times c' = \left( \frac{b \times c}{v}\right) . \left( \frac{c \times a}{v} \times \frac{a \times b}{v}\right) $ $ = v^{-1} \left(b \times c\right). \left[ \left(c \times...
  8. C

    MHB How to Find Cosine from Secant Using Trig Identities?

    If \cos(\pi/3)= \frac{1}{2}, find \sec(\pi-\pi/3) Someone really give me step-by-step explanation. I really don't know what identity to use, and no idea how to get cosine to secant. Please, it would help. I do have more questions if you help me dissect this problem. XD Thanks so much in advance!
  9. C

    MHB Trigonometric Identities Problem

    1) If \tan(\pi/4)=1, find \cot(\pi-\pi/4). 2) If \cot(17^{\circ}) = 3.2709, find \tan(73^{\circ}) 3) If \cot(\theta) = \frac{-9}{2} with \theta in Quadrant II, find \sin (\theta) --------------------------------------------- I really have no idea how to solve any of these problems. I have...
  10. C

    'Diracology'-some simple identities

    1. Homework Statement I can't get the latex to render so I use ##\tilde a## to mean the slashed notation commonly seen in the Dirac equation, so ##\tilde a = \gamma^{\nu}a_{\nu}## which is also ##a_{\nu}\gamma^{\nu}## I think. Prove the following: ##\gamma^{\mu} \tilde{a} \gamma_{\mu} = -2...
  11. C

    MHB NEW Beginner's Trigonometry Identities Problem

    Alright. I am sort of understanding this section on my online math lesson, but I am still struggling with it. Would be gladly appreciated if someone could help me with this: If cosΘ = -4/9 with Θ in Quadrant II, find sinΘ
  12. S

    MHB Which of the following identities is true. Justify your answer. n = O(4n ) 4 n = O(n)

    Which of the following identities are true. Justify your answer. a)$n! = O(4^n)$ b)$4^n = O(n!)$ I have NO clue what to do here. First I was thinking let $n = 0$ so that $1 = O(1)$ (constant time complexity?)
  13. M

    MHB Identities of Chebyshev polynomials

    Hey! :o We are given the polynomial functions $$T_0(x)=1, T_1(x)=x, x \in \mathbb{R} \\ T_{n+1}(x)=2xT_n(x)-T_{n-1}(x), n \in \mathbb{N}, x \in \mathbb{R}$$ (Chebyshev polynomials) Using induction I have to show that: the degree of $T_n$ is $n$ $\forall n \in \mathbb{N}$ : $T_n(1)=1$...
  14. C

    MHB Simple Trigonometric Identities

    If (sinΘ) = 2/3 with Θ in quadrant 1, find (secΘ)[/SIZE] Θ = theta Completely new at trigonometric identities, would be a great help!
  15. T

    How do Peskin/Schroeder derive 2-component Fierz identities?

    On page 51 Peskin and Schroeder are beginning to derive basic Fierz interchange relations using two-component right-handed spinors. They start by stating the trivial (but tedious) Pauli sigma identity...
  16. M

    Simplify the proof of different vector calculus identities

    Is there a way to simplify the proof of different vecot calculus identities, such as grad of f*g, which is expandable. And also curl of the curl of a field. Is there a more convenient way to go about proving these relations than to go through the long calculations of actually performing the curl...
  17. binbagsss

    Tricky Quadratic formula / trig identities

    Im to solve ##(k+l)^{2}e^{-ila}-(k-l)^{2}e^{ila}=0##, for ## k##, The solution is ##k=l(e^{ial}-1)/(e^{ial}+1)=il tan(al/2)## FIRST QUESTION So it's a quadratic in k, should be simple enough, my working so far using the quad. formula is ##k= (4l^{2}(e^{-ila}+e^{ila})\pm...
  18. Drafter

    The Derivation of Several Triginometric Identities

    I would like to know how can I derive the double, half, product to sum, and sum to product identities of trigonometry using simple algebraic means. And which books (which I prefer) should I pickup at the library on this subject to actually learn these derivations? Or at the very least some...
  19. earthloop

    Simplifying with trig identities

    Homework Statement [/B] Hi, I am currently working through a textbook, and the following simplification is given for an example question: I can't seem to work out how they have moved from cos(pi+n*pi) to cos(pi)cos(n*pi) so easily? Is there a simple trick I have missed? I understand the...
  20. S

    Intuitive interpretation of some vector-dif-calc identities

    Dear All, I am studying electrodynamics and I am trying hard to clearly understand each and every formula. By "understand" I mean that I can "truly see its meaning in front of my eyes". Generally, I am not satisfied only by being able to prove or derive certain formula algebraically; I want to...
  21. T

    I badly with general answers to trig equations. (Using identities)

    I'm completely lost here. I've got the cheat sheet of trig rules, but they don't appear to be helping me, I've watched a half dozen videos on each of cos sin and tan, and nearly all of them discuss the wrong topic. I don't want help on any single problem, but advice. How can I make sense of the...
  22. M

    Understanding Trigonometric Identities: Solving for -1

    Homework Statement Show that (sin^4 x + (sin^2 x * cos^2 x)) / (cos^2 x - 1) == -1 Homework Equations Sin^2 x + cos^2 x == 1 The Attempt at a Solution (sin ^4 x + (sin^2 x * cos^2 x)) / (cos^2 x - 1) = ((sin^2 x)(sin^2 x) + (sin^2 x * cos^2 x)) / (cos^2 x - 1) =((sin^2 x)(1 - cos^2 x) +...
  23. W

    Trigonometry identities and equations

    1) Question statement: Simplify 2sec^2x-2sec^2xsin^2x-sin^2x-cos^2x 2)Relevant equations: tan A=sinA/cos A 1+tan^2A=sec^A cot A=1/tanA cot A=cos A/sinA sin^2A+cos^2A=1 secA=1/cos A cosecA=1/sinA 1+cosec^2A= cot^2A sin2A=2sinAcosA cos2A=1-2sin^2A=cos^2A-sin^2A=2cos^A-1 tan2A=(2tanA)/1-tan^2A 3)...
  24. W

    Trigonometric Identities and equations

    1) Problem statement: Solve the trigonometric equation for the domain is. 0°<x<360° 5(sinx - cosx) = 4sinx - 3cosx 2) Relevent equations: secx=1/cosx cosec x=1/sinx cot x= 1/tanx tan x=sinx/cosx cot x=cosx/sinx sin^2 x + cos^2 x=1 1 + tan^2 x= sec^2 x 1+ cot^2 x = cosec^2 x Template of answer...
  25. 1

    Vector Calculus - Use of Identities

    Homework Statement By using a suitable vector identity for ∇ × (φA), where φ(r) is a scalar field and A(r) is a vector field, show that ∇ × (φ∇φ) = 0, where φ(r) is any scalar field. Homework Equations ∇×(φA) = (∇φ)×A+φ(∇×A)? The Attempt at a Solution I honestly have no idea how to even...
  26. D

    MHB Trigonometric identities transformation last one

    Transform the left hand member into the right hand member. $\frac{\tan\alpha+\tan\beta}{\sec\alpha-\sec\beta}=\frac{\sec\alpha+\sec\beta}{\tan\alpha-\tan\beta}$By using cross multiplication I was able to prove this identity but what I actually want to accomplush is to transform the left member...
  27. D

    MHB Trigonometric identities transformation

    I already did everything that I can to transform the left side member to the right side member but I always get a jumbled terms. Please give me a hand on this problem. $(2\sin^{2}(\theta)-\cos^{2}(\theta))^{2}-9(2\sin^{2}(\theta)-1)^{2}=(2-3\sin^{2}(\theta))(2+3\sin(\theta))(3\sin(\theta)-2)$
  28. D

    MHB Transforming Trigonometeric Identities II

    Transform the left member to the right member. $\frac{\left(\sin^{2}(\phi)\cos^{2}(\phi)+\cos^{4}(\phi)+2\cos^{2}(\phi)+\sin^{2}(\phi)\right)}{1-\tan^{2}(\phi)}=\frac{3+\tan^{2}(\phi)}{1-\tan^{4}(\phi)}$ I begin by regrouping the numerator of the left...
  29. A

    MHB Algebraic Proofs and Verifying identities

    Hey all! I am having some trouble with a certain problem on my homework. I would like some guidance. I have to prove one side of the equation is equal to the other, as you may know, as this is an algebraic proof. This in itself isn't too hard. The hard part is just this one particular problem. I...
  30. D

    MHB Can you prove this identity using trigonometric identities?

    Help me get started with these problems. Prove the following$\frac{\tan(A)}{1-\cot(A)}+\frac{\cot(A)}{1-\tan(A)}=\sec(A)\csc(A)+1$ $\frac{\sec(A)-\tan(A)}{\sec(A)+\tan(A)}=1-2\sec(A)\tan(A)+2\tan^{2}(A)$
  31. paulmdrdo1

    MHB Solving Proving Identities - Trig Chapter Problems

    Hey guys, I'm about to finish answering the chapter problems on my trigonometry books about proving Identities, there are 2 problems that made me scratch my head though. They are the last 2 problems left that I'm not yet able to verify. I would greatly appreciate it if you could lend me some...
  32. paulmdrdo1

    MHB Proving Identity 46: $\cos^{6}(A)+\sin^{6}(A)=1-3\sin^{2}(A)\cos^{2}(A)$

    Again I'm stuck with another problem in proving identities. This is the 46th item in the list of 53 identities that I'm asked to verify and so far I was able to prove 45 of them, there are 8 items left and one of them is this Identity $\cos^{6}(A)+\sin^{6}(A)=1-3\sin^{2}(A)\cos^{2}(A)$ Thanks!
  33. D

    MHB Establishing Identities - Possible Misprint on Worksheet

    Hey guys, I'm just going over my work files for this week and I've noticed 2 problems that have fooled me, and seem a little bizarre for my current mathematical level. They do involve trig, and a part of me is hoping for a misprint on the sheet, but I'm more sure that it's my own inadequacies...
  34. evinda

    MHB You're welcome! Glad I could help. (Thumbs up)

    Hi! (Wave) Let $R$ be a relation. Show the following sentences: $dom(R^{-1})=rng(R)$ $rng(R^{-1})=dom(R)$ $fld(R^{-1})=fld(R)$ $(R^{-1})^{-1}=R$ That's what I have tried: Let $x \in dom(R^{-1})$. Then $\exists y$ such that $<x,y> \in R^{-1} \Rightarrow <y,x> \in R \Rightarrow x \in...
  35. S

    Dirac delta function identities

    hi deoes anyone know any online resource for proofs of Dirac delta function identities and confirming which representations are indeed DD functions Thanks a lot.
  36. evinda

    MHB Prove Identities of Sets Hello! (Wave)

    Hello! (Wave) Let $U$ be a set and $A,B$ subsets of $U$. I want to prove the following identities: $A \cap A^c= \varnothing, A \cup A^c=U$ $(A^c)^c=A$ $(A \cap B)^c=A^c \cup B^c$ $(A \cup B)^c=A^c \cap B^c$ $A \setminus B=A \cap B^c$ That's what I have tried: Let $x \in A \cap...
  37. D

    MHB Finding Exact Value using Trig Identities and Complementary Angle Theorem

    Hey guys, I've been trying to wrap my mind around this problem but I've really come up short. Any help would be amazing. If tanX=10 Find the exact value of cot(pi/2 - x)
  38. B

    Proving Vector Calculus Identities: Tips and Tricks

    Homework Statement div(øu) = ødivu + ugradø Homework Equations divergence of scalar field = f,ii divergence of vector field = ui,i The Attempt at a Solution I've heard this is a simple proof, but this is my first one of 8 or so proofs I need to complete for homework, and I'm...
  39. D

    MHB What is the equivalent identity for $\sin^2\omega t$?

    is there an equivalent identity for $\sin^2\omega t?$ please tell me.REGARDS!
  40. KleZMeR

    Simplify with Trigonometric Identities

    Homework Statement I'm trying to simplify some trigonometric expressions, I'm attaching my work here. This comes from a famous physics problem i.e. the rod with two masses spinning on a circle. I've tried many times but I just can't get it. Any help on which identities to use would really...
  41. K

    Understanding Quantum Physics Identities

    Hey, so today for our quantum physics class we were supposed to go through these identities, |+_x > = \frac{1}{2^{0.5}} (|+> + |->)|-_x > = \frac{1}{2^{0.5}} (|+> - |->) |+_y > = \frac{1}{2^{0.5}} (|+> + i|->) |-_y > = \frac{1}{2^{0.5}} (|+> - i|->)where |+ (x)> would represent spin up...
  42. C

    Need help proving some trig identities

    Proving identities is a pain! Thanks in advance, guys! Homework Statement 1. 1 + sec^(2)xsin^(2)x = sec^(2)x 2. sinx/1-cosx + sinx/1+cosx = 2cscxHomework Equations The Attempt at a Solution For the first problem, this is the best I got: 1 + sec^(2)x(1-cos(2)x) For the second problem, I...
  43. D

    Algebric identities homework

    Homework Statement Factorate the expression a(1-b²)(1-c²) + b(1-c²)(1-a²) + c(1-a²)(1-b²) - 4abc Homework Equations Algebric identities The Attempt at a Solution It seems to be an olympic problem, and I can't find the right factors for it. I would really appreciate the help and, if...
  44. Greg Bernhardt

    What are trigonometric identities

    Definition/Summary In a right-angled triangle, with a hypotenuse ("hyp"), and with sides adjacent ("adj") and opposite ("opp") to the acute angle we are interested in, the six basic functions are defined as follows: sin = opp/hyp, cos = adj/hyp, tan = opp/adj, cosec = 1/sin, sec = 1/cos...
  45. C

    Correlation functions and Ward identities

    The definition I have for a 2-point correlator is $$\langle \phi_1(x_1)\phi_2(x_2)\rangle = \frac{1}{Z} \int \mathcal D \phi \,\,\phi_1(x_1)\phi_2(x_2) \exp-S[\phi],$$ where ##Z = \int \mathcal D \phi \,\,\exp-S[\phi]##. I am trying to understand the overall physical meaning of such a quantity...
  46. Dethrone

    Understanding integration with trig identities, and absolute value

    Homework Statement In integration, we are allowed to use identities such as sinx = \sqrt{1-cos^2x}. Why does that work, and why doesn't make a difference in integration? Graphing \sqrt{1-cos^2x} is only equal to sinx on certain intervals such as(0, \pi) and (2\pi, 3\pi). More correctly...
  47. Dethrone

    MHB Integration with trig identities and absolute value

    In integration, we are allowed to use identities such as sinx = \sqrt{1-cos^2x}. Why does that work, and why doesn't make a difference in integration? Graphing \sqrt{1-cos^2x} is only equal to sinx on certain intervals such as (0, \pi) and (2\pi, 3\pi). More correctly, shouldn't we use the...
  48. D

    MHB Proving identities using demoivre's formula

    prove the following identities $\cos(2\theta)=\cos^{2}(\theta)-\sin^{2}(\theta)$ $\sin(2\theta)=2\cos(\theta)\sin(\theta)$our Demoivre's formula says $z=r\left(\cos(\theta)+i\sin(\theta)\right)$ I don't know how use it to prove the identities above please help me get started. regards!
  49. E

    Levi civita symbol and kronecker delta identities in 4 dimensions

    I'm trying to explicitly show that \varepsilon^{0 i j k} \varepsilon_{0 i j l} = - 2 \delta^k_l I sort of went off the deep end and tried to express everything instead of using snazzy tricks and ended up with \begin{eqnarray*} \delta^{\mu \rho}_{\nu \sigma} & = & \delta^{\mu}_{\nu}...
  50. L

    Trig identities and complex numbers help.

    Please forgive me as I may have to edit this post to get the equations to show properly. I am doing some work with AC circuits and part of one of my phasor equations has this in it: \frac {2i} {1+cos(θ) + i sin(θ)} - i , where i is the imaginary number \sqrt{-1}. However, knowing the...
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