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

    Verifying Trigonometric Identities

    I have been having lots of trouble verifying trigonometric identities. I know the fundamental identities but I am actually having trouble with the algebra that goes along with the problems. The problem I am working on now is: cos(x)-tan(x)/sin(x)cos(x) = csc^2 (x) - sec^2 (x) (The csc...
  2. M

    Trig' Identities - Addition Formulae

    Homework Statement Given that cos C = 12/13 where C is a reflex and sin D = 3/5 where D is acute, find the exact value of cos ( C + D ). Homework Equations The Attempt at a Solution I used the Addition Formulae: cos(A+B) = cosAcosB-sinAsinB cos(12/13)cos(2/5) -...
  3. Z

    What is the Range of y in (y+1)/(y-1)=sin2x/sin2a?

    Homework Statement I got this expression while solving a problem. (y+1)/(y-1)=sin2x/sin2a we need to find the range of y Homework Equations The Attempt at a Solution here y=(sin2x+sin2a)/(sin2x-sin2a) Numerator of RHS lies between sin2a-1 and 1+sin2a Denominator lies...
  4. J

    Calc- prove sinh(x) and cosh(x) identities?

    Homework Statement Prove cosh(2x)= cosh^2(x) + sinh^2(x) Homework Equations sinh(x) ≡ [ e^(x) - e^(-x) ] / 2 cosh(x) ≡ [ e^(x) + e^(-x) ] / 2 The Attempt at a Solution = { [ e^(x) + e^(-x) ] / 2 }² + { [ e^(x) - e^(-x) ] / 2 }² = [ e^(x) + e^(-x) ]² / 2² + [...
  5. S

    Simplifying Trigonometric Expressions

    Homework Statement (Sin 2θ / sinθ) - (cos 2θ/ cos θ) = Sec θ just trying to match one side to the other Homework Equations all trig identities The Attempt at a Solution broke down sin2θ into 2sinθ cosθ then reduced the sinθ in the denominator giving me sinθ cosθ -...
  6. E

    Caculus Help : Integrating with trig identities?

    Homework Statement integrate: sin (2x)/(1+sinx) Homework Equations (sin x)^2 + (cos x) ^2 = 1 sin (2x) = 2 sin x cos x cos (2x) = (cos x)^2 - (sin x)^2 The Attempt at a Solution I've been trying to integrate this thing for about an hour by rearranging various trig...
  7. F

    Is (B.\nabla)A the same as B(\nabla.A)?

    Homework Statement Prove the following vector identity: \nablax(AxB) = (B.\nabla)A - (A.\nabla)B + A(\nabla.B) - B(\nabla.A) Where A and B are vector fields. Homework Equations Curl, divergence, gradient The Attempt at a Solution I think I know how to do this: I have to...
  8. S

    Proving Trigonometric Identities

    Homework Statement Prove (using the left side): sinΘ tanΘ = cosΘ sec^2Θ - cosΘ Homework Equations The Attempt at a Solution
  9. Y

    Question about function defined in a region using Green's identities.

    I want to verify my understand of this. Let u defined in region \Omega with boundary \Gamma. If u = 0 \hbox { on the boundary } \Gamma, then u = 0 \hbox { in the region } \Omega. The way to look at this, suppose u is function of x component called Xand y component called Y. So...
  10. Y

    Help solving Green's identities question.

    Homework Statement Suppose u is harmonic (\nabla^2 u = 0 ) and v=0 \;\hbox{ on } \;\Gamma where \Gamma is the boundary of a simple or multiply connected region and \Omega is the region bounded by \Gamma. Using Green's identities, show: \int \int_{\Omega} \nabla u \cdot \nabla v \...
  11. Y

    Solving Dirichlet problem using Green's identities.

    This is to solve Dirichlet problem using Green's identities. The book gave some examples. My question is: Why the book keep talking v is harmonic(periodic) function. What is the difference whether v is harmonic function or not as long as v has continuous first and second derivatives...
  12. S

    Linear algebra identities of inverse matricies

    Homework Statement Left Inversion in Rectangular Cases. Let A^{-1}_{left} = (A^{T}A)^{-1}A^{T} show A^{-1}_{left}A = I. This matrix is called the left-inverse of A and it can be shown that if A \in R^{m x n} such that A has a pivot in every column then the left inverse exists. Right...
  13. X

    Hyperbolic Function identities

    I have always been curious as to where the definition of cosh(x) and sinh(x) come from and how they are related to the natural exponential. I know it has something to do with Euler's formula but I don't know the details of the derivation. Could anyone shed some light on this? I haven't yet...
  14. S

    Linear algebra identities of inverse matricies/transpose

    Homework Statement Left Inversion in Rectangular Cases. Let A^{-1}_{left} = (A^{T}A)^{-1}A^{T} show A^{-1}_{left}A = I. This matrix is called the left-inverse of A and it can be shown that if A \in R^{m x n} such that A has a pivot in every column then the left inverse exists. Right...
  15. X

    How Can You Factor the Trigonometric Expression sin^3(x)-cos^3(x)?

    Homework Statement sin^3(x)-cos^3(x) sin(x) - cos(x) equals 1 + sin(x) + cos(x) Homework Equations Not sure :/ The Attempt at a Solution Not sure where to even start.
  16. A

    Trigonometric identities problem

    Given sinθ = 0.6, calculate tanθ without using the inverse sine function, but instead by using one or more trigonometric identities. You will find two possible values. I found one of the values using sin^2 (theta) + cos^2 (theta) = 1 I tried using cos (90 + theta)= sin theta to find the...
  17. S

    Set Theory Identities: A = B if A, B, and C satisfy key set relations

    Homework Statement Can you conclude that A = B if A, B, and C are sets such that A \cup C = B \cup C and A \cap C = B \cap C Homework Equations The above is part c of a problem. The problems a and b are as follows A) A \cup C = B \cup C My answer: I gave a counter example...
  18. R

    Verification of tensor identities

    Homework Statement Show that in 2 dimensions a skew-symmetric tensor of second rank is a pseudoscalar and that one of third rank is impossible. The Attempt at a Solution A11=A22=0, while A12=-A21, which makes A= A12+A21, which is certainly skew-symmetric, though I am not sure it is...
  19. Q

    Proving vector identities using Cartesian tensor notation

    Homework Statement 1. Establish the vector identity \nabla . (B x A) = (\nabla x A).B - A.(\nabla x B) 2. Calculate the partial derivative with respect to x_{k} of the quadratic form A_{rs}x_{r}x_{s} with the A_{rs} all constant, i.e. calculate A_{rs}x_{r}x_{s,k} Homework Equations The...
  20. S

    How can I prove the trig identity sinxcosxsec^2x = tanx?

    [b]1. How do I prove sinxcosxsec^2x=tanx Homework Equations sec^2x = tan^2x + 1 The Attempt at a Solution sinxcosx(tan^2x + 1) tan^2xsinxcosx +sinxcosx sin^2x/cos^2x*sinxcosx + sinxcosx - is this valid? I'm not sure what I've done is even correct - but it doesn't seem...
  21. D

    Do these curious identities hold true?

    These identities might be considered as true: \begin{array}{rcl} 1 - 1 + 1 - 1 + \cdots & = & \frac{1}{2} \\ 1 - 2 + 3 - 4 + \cdots & = & \frac{1}{4} \\ 1 - 4 + 9 - 16 + \cdots & = & 0 \\ 1 + 1 + 1 + 1 + \cdots & = & -\frac{1}{2} \\ 1 + 2 + 3 + 4 + \cdots & = & -\frac{1}{12} \\...
  22. M

    Vector Calculus Identities: Proving v · ∇v = ∇(0.5v2 + c × v)

    Homework Statement Show that v\nablav = \nablaxvxv v · ∇v = ∇(0.5v2 + c × v c=∇ × v My attempt ∇(A · B)= B · ∇A + A · ∇B + B×(∇×A) + A×(∇×B) Replace A and B with V ∇(v · v)= v · ∇v + v · ∇v + v×(∇×v) + v×(∇×v) v · ∇v = ∇(0.5v2 - v×(∇×v) Is v×(∇×v) = =∇ × v × v? And...
  23. A

    Why Can't I Simplify This Trigonometric Equation?

    Homework Statement 1/cscx-sinx = secx tanx Homework Equations cscx = 1/sinx secx = 1/cosx The Attempt at a Solution 1/cscx-sinx = secx tanx L.S. = 1/cscx-sinx = 1/(1/sinx)-sinx R.S. = secx tanx = (1/cosx)(sinx/cosx) This is where I'm getting confused. Why can't...
  24. N

    Trigonometry Identities: Simplifying Higher Powers

    1. sec^4 x + sec^2 x tan^2 x - 2 tan^4 x = ? The possible answers are: a. 4 sec^2 x b. 3 sec^2 x - 2 c. sec^2 x + 2 d. tan^2 x - 1 Homework Equations No idea. The Attempt at a Solution I'm not sure where to begin here. My book first doesn't cover anything above...
  25. J

    Solving Trig Identities: From \sqrt{2+2cosx} to 2cos(x/2)

    Homework Statement How to get from \sqrt{2+2cosx} to 2cos(x/2) The Attempt at a Solution I'm pretty much stuck on this. I can only see that is simplifies by taking out the common factor 2, which isn't correct.
  26. G

    Verifying Trig Identities: csc(A-B)=secB

    Homework Statement Verify that each equation is an identity- directions Problem- csc(A-B)=secB ---------------- <<< divide bar sinA-cosAtanB Homework Equations well i tried to put in terms of sin cos and I've gotten stuck The...
  27. S

    Proving vector identities with index notation (help with the del operator)

    Homework Statement Prove the vector identity: \left(a\times\nabla\right)\bullet\left(u \times v\right)=\left(a \bullet u \right)\left(\nabla \bullet v \right)+\left(v \bullet \nabla \right)\left(a \bullet u \right)-\left(a \bullet v \right)\left(\nabla \bullet u \right)-\left(u...
  28. P

    Solve Trig Identities Homework Equations

    Homework Statement 1. cos θ - 1 / cos θ = sec θ - sec^2 θ 2. (sin θ + cos θ)^2 = 1 + 2 sin θ cos θ 3. (1 / 1 + cos θ) + (1 / 1 - cos θ) = 2csc^2θ Homework Equations 1 / cos θ = sec θ 1 / sin θ = csc θ sin^2 θ + cos^2 θ = 1 cos^2 θ = 1 - sin^2 θ sin^2 θ = 1 - cos^2 θ...
  29. S

    Is cos(x - (pi/2)) equal to cos(x)tan(x)?

    cos(x-(pie/2))=cos(x)tan(x) I have to verify this identity and can't seem to figure it out. cos (x-y)=cos x * cos y + sin x * sin y well since cos y = 0, it kind of eliminates that side of the equation and I end up with sinx * sin1 Did I go about this all wrong?
  30. P

    Proving Trig Identities: Is this Question Referring to the Pythagorean Identity?

    Homework Statement Prove Trig. Identities 1. sec θ (sec θ - cos θ)= tan^2 θ Homework Equations sec θ = 1/cos θ tan θ = sin θ/ cos θ cot θ = cos θ / sin θ The Attempt at a Solution 1. sec θ * sec θ - sec θ * cos θ 1/ cos θ * 1/ cos θ - 1/ cos θ * cos θ ----> cos θ is...
  31. I

    Single Trigonometric Functions ( trig identities)

    Homework Statement Cos^2x-Sin^2x/2 SinxCosx The Attempt at a Solution I changed cos^2x to 1- sin^2x which then the equation was 1- s sin^2x/2snxcosx and i have no idea how to make this a single trig. function
  32. N

    Trig identities: does this equal zero?

    Homework Statement This is actually many steps through a calculus problem involving trig functions. I have not included the problem because I'm trying very hard to figure it out on my own (at least as far as it's possible). I've found the answer I'm looking for, but it's attached to a bunch...
  33. C

    How to Prove a Trigonometric Identity

    cos = sin (pi/2-theta)
  34. K

    Trigonometric Identities

    Homework Statement Without using tables(calculators) find the numerical value of Sin[Pi/8]^2 - Cos[3 (Pi/8)]^4 Homework Equations The Attempt at a Solution I tried changing it to: 1-cos[pi/8]^2 - cos[3pi/8]^4 but have no idea where to go... its really got me scratching my...
  35. silvermane

    Combinatorial Proofs of Binomial Identities

    Homework Statement (Give a combinatorial proof of each of the following identities. In other words, describe a collection of combinatorial objects and then explain two different methods for counting those objects. Leave each identity in the form given. Do not rearrange terms or use any other...
  36. P

    Trigonometric Identities Section

    Homework Statement 3sinx = 1 + cos 2x Homework Equations N/A The Attempt at a Solution 3sinx = 1+1 - 2sin2x (Trigonometric Identity) 3sinx = 2-2sin2x 2sin2x + 3sinx = 2 I do not know where to go from there. My book tells me the answer is 30o and 150o
  37. X

    Pythagorean Identities and Equations

    Homework Statement 12cot2(2x)=4 (8 exact answers) 3cos2(3x)=2sin(3x) (6 decimal answers) cos(3x)=sec(3x)+2tan2(3x) (6 exact answers) hint(get all cosines) <- teacher wrote that And then I have to prove by working on one side: sin3(x) / 1-cosx = sinx*cosx+sinx and (sinx / cosx-1) + (sinx /...
  38. K

    Simplification Through Fundemental Identities

    Homework Statement (sin3+cos3)/(sin+cos) Homework Equations sin2+cos2=1 1 +cot2=csc2 1+tan2=sec2 Are these sufficient? The Attempt at a Solution Confusion. Epic confusion. I might have had it yesterday, but the mathematical equivalent of a writers anti-tank roadblock. At best...
  39. D

    Trig Identities Question NEED HELP

    How do I prove that (1+sin2x) = (cosx) (cosx + sinx) (cosx - sinx)
  40. C

    How Do You Prove These Trigonometric Identities?

    Can someone please help me with these two questions. Th first one is prove: 1-tan^2x ________ = cos2x 1+tan^2x & the second one is prove: sinx+ sinxcot^2 = secx
  41. S

    Help with identities math problem?

    The question asks you to simplify sin(3pi/2+x). I know that you've got to use the sin(a+b)=sinacosb+cosasinb but I'm not sure how to solve it when it's not a special triangle. Any help is much appreciated.
  42. H

    What are some useful vector calculus identities for vector fields A and B?

    Homework Statement For arbitrary vector fields A and B show that: ∇.(A ∧ B) = B.(∇∧A) - A.(∇∧B) The Attempt at a Solution I considered only the 'i'-axis, by saying that it is perpendicular with A and B and then I expanded both the left and right side out. The working is...
  43. D

    Identities simplify expression

    I have been trying to do this problem for a long time, and still can not do it. I know the answer is sin2x, but I have no idea how to do it: write expression as sine, cosine, or tangent of an angle sin3xcosx - cos3xsinx THANKS!
  44. D

    Trig identities help How to simplify by multiplication/division

    Ok guys, here's my problem. I left on vacation with my parents before I learned how to do these correctly. I have been trying and I sort of have the gist of them down. For instance, tan A*sec A simplified is sin A. A=theta. But as I move on they stop making sense, and this is where my problem...
  45. M

    Trigonometric Identities algebra

    hello. i have a question about trignometric identities.. it's realtivly easy, but am struggling with the algebra in it ( Algebra + trig = Very confusing to me ) Prove that .. [Sinx/(1+Cosx)] + [(1+cosx) / sinx] = 2csc x i manged to get it to [Sin2x+1+2cosx+cos2x] / SinxCosx...
  46. S

    What is the fundamental identity used to prove csc2α - 1 = cos2α / csc2α?

    Homework Statement Okay, so this is some trig I learned last year but have since forgotten. If you can give me the first step, I can solve the rest on my own. The given statement is true and you have to prove why using Pythagorean Identities. csc2\alpha-1 = cos2\alpha ________ csc2\alpha
  47. G

    Ward identities in Minimal Subtraction Scheme

    In QED, the Ward identities set Z_1=Z_2 and Z_1 - 1 = \left. {\frac{{d\Sigma \left( p \right)}}{{dp}}} \right|_{p = m} . This can be shown explicitly for the 1-loop calculations if one uses an on-shell subtraction scheme, where the renormalized mass and charge are identical to the...
  48. I

    Unlocking Trig Identities: Proving sin(2x + pi/3)

    Homework Statement By using trig formulas show that, sin(2x + pi/3) = sin(2x) + sin(2(x+pi/3)) Homework Equations Trig Identities The Attempt at a Solution I've used double angle formulas, sin(a+b) formulas, I just can't seem to get it.
  49. A

    Real life application of trignometry identities

    Homework Statement Can anyone here help me find how and where trig identities are used in real life. can you also tell me which identity(with example) and is used for what purpose and how. Homework Equations The Attempt at a Solution I found out that it is used in the field...
  50. K

    Using Trig Identities to see if derrivatives are equal

    Homework Statement for homework we have to find 2 different dirrivatives of the same problem (one may be incorrect) and then tell if they are equal 2. The attempt at a solution original y=sec(x)*cot(x) the two derrivatives (i know these are correct becase i have compared with others in...
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