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. P

    Trigonometric identities and complex numbers

    Homework Statement Show, using complex numbers, that sin(x)+cos(x)=(√2)cos(x-∏/4) Homework Equations cos(x)=(e^(ix)+e^(-ix))/2 sin(x)=(e^(ix)-e^(-ix))/2i e^ix=cos(x)+isin(x) The Attempt at a Solution I was given the hint that sin(x)=Re(-ie^(ix)), but have thus far not been...
  2. S

    Solve Math Trig Identities w/ Sin & Cos Only

    Hi I am in need of some help for this question: 1+tanx/1-tanx = tan(x+(∏/4)) It is easy to solve with the tan trig identites on the right side however, my teacher had told me to do it with SIN and COS only. I am not sure if its possible and was looking for some insight Left Side...
  3. W

    Help with Double Angle Identities

    Help with Double Angle Identities! cos θ = 24/25 The angle lies in quadrant 1; 0<θ<90 Find sin2θ I know you would use either the formula cos^2θ - sin^2θ or 2sinθcosθ And I know that the answer is 339/625, but I do not know how to get that answer?
  4. T

    Solving Equations Using Trigonometric Identities

    Hey PF! Homework Statement Find exact solutions for the following equations over the domain 0 ≤ x <2π 2sinx = 3 + 2cscx Homework Equations sin2+cos2=1 The Attempt at a Solution 2sinx = 3 + 2cscx 2sinx = 3 +2(1/sinx) sinx = 3/2 + 1/sinx sinx - 1/sinx = 3/2 (1-1-cos2x)/sinx = 3/2 -cos2x/sinx =...
  5. D

    MHB Is the Bottom Equality in this Complex Mathematical Equation Correct?

    I have another answer to this but I believe this one is correct. I need someone else to check it out since I have been looking at it too long. Is the bottom equality correct? \begin{alignat*}{3} \frac{\partial^2}{\partial t^2}x_1 + x_1 & = & F\cos t - 2[-A'\sin t + B'\cos t] - c[-A\sin t +...
  6. M

    Using Determinant Identities to solve

    Sorry for the format, I'm on my phone. Lets say the matrix is | 1 1 1 | | a b c | | a^2 b^2 c^| Or {{1,1,1} , {a, b,c} , {a^2, b^2,c^2}} Show that it equals to (b-c)(c-a)(a-b) I did the determinant and my answer was (bc^2) - (ba^2) - (ac^2) + (ab^2) + (ca^2) - (cb^2)...
  7. E

    Integration by parts, more than one variable, and green's identities etc.

    I was wondering if someone can show me or point me to a worked out example using integration by parts for more than one variable (as used in relation to pde's, for example). While I took pdes and calc 3, its been awhile and I don't know if I ever understood how to work out a concrete example...
  8. N

    Trig half-angle identities?

    Apparently our professor expects us to know these half-angle identities (http://www.purplemath.com/modules/idents.htm) Without going through them in class or us learning them in high school.. Can somebody explain how these were derived? Does the derivation come from the angle-sum and...
  9. J

    Integrating cot^4 x (csc^4 x) dx Using Identities and U Substitution

    ∫(cot^4 x) (csc^4 x) dx Wolfram wants to use the reduction formula, but I'm meant to do this just using identities and u substitution. I was thinking something along the lines of: =∫cot^4 x (cot^2 x + 1)^2 dx =∫cot^8 x + 2cot^6 x + cot^4 x dx but I don't know where to go from there.
  10. J

    Prove Quadruple Product Identity from Triple Product Identities

    Homework Statement I need to prove the identity: (a×b)\cdot(c×d)= (a\cdotc)(b\cdotd)-(a\cdotd)(b\cdotc) using the properties of the vector and triple products: Homework Equations a×(b×c)=b(a\cdotc)-c(a\cdotb) a\cdot(b×c)=c\cdot(a×b)=b\cdot(c×a) The Attempt at a Solution I...
  11. B

    Limits Involving Trigonometric Functions (identities)

    Does anyone know of websites where I can find many problems on the topic in the title line (my textbook has far too few)? Thanks!
  12. S

    Integration Using U-Substitution involving Trig Functions and Identities

    1.) ∫[(7 sin (x))/[1+cos^2(x)]] * dx 2.) I'm looking at the trig identity sin^2 x+cos^2 x=1, and am wondering if I could use that in solving the problem. Or should I use u=sin x, then du= cos x, then plug those in? 3.) so I thought maybe it would be easier to separate the two...
  13. D

    MHB Trig identities Fourier Analysis

    Prove the identities $$ \frac{\sin\left(\frac{n + 1}{2}\theta\right)}{\sin\frac{\theta}{2}}\cos\frac{n}{2}\theta = \frac{1}{2} + \frac{\sin\left(n + \frac{1}{2}\right)\theta}{2\sin\frac{\theta}{2}} $$ By using the identity $\sin\alpha + beta$, I was able to obtain the $1/2$ but now I am not to...
  14. A

    A question about trigonometric identities

    Is this identity possible? cot 2x = \frac{cos3x + cosx}{sin 3x + sinx} Thanks!
  15. A

    Proving Trigonometric Identities problem

    Homework Statement Verify that \frac{cosθ}{1-tanθ} + \frac{sinθ}{1-cotθ} = sinθ + cosθ is an identity.Homework Equations - Reciprocal Trigonometric Identities - Pythagorean Trig IdentitiesThe Attempt at a Solution Every time I try to manipulate the LHS of the equation I always get -1 and as far...
  16. A

    Proving Trigonometric Identities

    Homework Statement Prove that sin6 + cos6 = 1 - 3sin2cos2 Homework Equations (1) The Attempt at a Solution I tried to convert those all in terms of sine and cosine only but it didn't work.
  17. C

    Proving trig identities with dot and cross products

    Homework Statement The two vectors a and b lie in the xy plane and make angles α and β with the x axis. a)By evaluating a • b in two ways (Namely a •b = abcos(θ) and a • b = a1b1+a2b2) prove the well-known trig identity cos(α-β)=cos(α)cos(β)+sin(α)sin(β) b)By similarly evaluating...
  18. D

    Proving trigonometric identities

    Homework Statement Prove that: (1-tanθ)/(1+tanθ)=(cotθ-1)/(cotθ+1) Homework Equations Trig Identities: tanθ= sinθ/cosθ cotθ= cosθ/sinθ 1+tanθ=secθ 1+cotθ=cosecθ The Attempt at a Solution These sorts of equations are coming up a lot and I am having trouble understanding...
  19. N

    Help spotting trig identities to simplify integration

    Hello, Say I'm working with ∫ sqrt(1-cos(t)) dt I end up with a substitution of u = 1-cos(t) and dt = du/sin(t) sub back in: ∫ sqrt(u) / sin(t) du Still got a t in there ... hrrmmm So I go to wolfram alpha for some inspiration and 'show steps'...
  20. D

    Trigonometric identities hard question

    Homework Statement Simplify the following: sin(b)/cos(b) + cos(b)/sin(b) Homework Equations Trigonometric identities The Attempt at a Solution Ok so i have no clue how to do this,I keep trying but can't seem to get the right answer, I have tried to do this: sin(b)/cos(b) +...
  21. P

    Proof of difference identities for cosine

    Hi, I am working on proofs of the difference identities for sine, cosine, and tangent. I am hoping to solve these using a specific diagram (attached). I was wondering if you could help me with the difference of cosines. Is it possible to derive it using the attached diagram? If so, how...
  22. B

    Gradient and Divergent Identities

    Homework Statement I need to show that ##\displaystyle\int_\Omega (\nabla G)w dxdy=-\int_\Omega (\nabla w) G dxdy+\int_\Gamma \hat{n} w G ds## given ##\displaystyle \int_\Omega \nabla F dxdy=\oint_\Gamma \hat{n} F ds## where ##\Omega## and ##\Gamma## are the domain and boundary respectively...
  23. F

    Question about Trig Identities

    Homework Statement This isn't really a problem that was assigned to me, (I'm studying independently) I just have a question about the general concept behind some identities. Homework Equations sin(a+b) = sin(a)*cos(b) + cos(a)*sin(b) sin(theta) = sin(180-theta) The...
  24. P

    Multiplicative and additive identities as successors

    Fact: The ring of integers Z is totally ordered: for any distinct elements a and b in Z, either a>b or a<b. Fact: The ring of integers is discrete, in the sense that for any element a in Z, there exists an element b in Z such that there is no element c in Z with a<c<b, and the same argument...
  25. H

    MHB Can Trig Identities be Derived from Easier Formulas?

    I know you can derive the double angle formulas for sin(2a) and cos(2a) from Euler's identity, but is there any way to derive the tan(2a) in a similar manner from an easier formula? What about the addition/subtraction formulas (i.e. sin(a+b), etc.)
  26. 4

    Arctan Identities Via Exponentiation

    Is it possible to prove identities involving arctan by complex exponentiation? I had in mind something like the following for the arctan angle addition formula, but I feel there is something not quite right in the argument. $$\arctan{(a)}+\arctan{(b)}=...
  27. A

    Problem needing trig identities to find exact value

    Homework Statement Find the exact value of: sin (-5∏/12) 2. The attempt at a solution sin (-45° + -30°) = sin -45° cos -30° + cos -45° sin -30° = (sqrt (2) / 2 )(sqrt (3) / 2 ) + (sqrt (2) / 2)(1 / 2) = (sqrt (6) + sqrt (2)) / 4 However, the book has (-sqrt (6) -...
  28. K

    Algebraic proofs of trigonometric identities

    Hello all, I was wondering if someone has ever found a purely algebraic proof for the addition/subtraction theorems of trigonometry, mainly sin(a+b)=sin(a)cos(b)+sin(b)cos(a). Given a right triangle: Let x be one of the perpendicular legs and let the other leg be composed of two parts, y1...
  29. B

    Prove Calculus Identities: f, g Real Valued Functions

    Homework Statement Suppose f is a continously differentiable real valued function on R^3 and F is a continously differentiable vector field Prove 1)##\oint (f \nabla g +g\nabla f) \cdot dr=0## 2) ##\oint(f \nabla f)\cdot dr=0##Homework Equations ##\nabla f = f_z i+ f_y j+f_z k## Real valued...
  30. miraiw

    Identities of nested set algebraic expressions

    Are there any useful identities for simplifying an expression of the form: $$((\ldots((x_{1} *_{1} x_{2}) *_{2} x_{3}) \ldots) *_{n - 1} x_{n})$$ Where each $$*_{i}$$ is one of $$\cap, \cup$$ and $$x_1 \ldots x_n$$ are sets? I believe I found two; though I haven't proved them, I think they...
  31. N

    Use trig identities to simplify an expression (has sins and cosines)

    Homework Statement Use fundamental identities to simplify the expression: (sinx)^2 - (cosx)^2 ____________________ (sinx)^2 - (sinx cosx)*note: it's a numerator and denominator. The underscore line is the fraction line. *note: The answer in the back of the book is "1 + cotx" but I would...
  32. binbagsss

    Is 2a/sin 2x Equivalent to a Cot x?

    Is 2a/sin 2x equivalent to a cot x?
  33. A

    Trig Identities Applications Question

    Hello! I've been tackling the question 'Express sin3x+sinx as a product and hence solve 1/2(sin3x+sinx)=sin2x ; x∈R' but I'm stumped - I'm not sure whether I've even approached it correctly. This is what I did: sin(3x+x)=sin3x.cosx+sinx.cos3x inserting this into the second equation...
  34. T

    Congruence identities using Fermat's Little Theorem

    Homework Statement Show the remainder when 43^43 is divided by 17. Homework Equations $$43 = 16 \times 2 + 11$$ $$a^{p-1}\equiv1\ (mod\ p)$$ The Attempt at a Solution I believe I can state at the outset that as: $$43\equiv9\ (mod\ 17)$$ Then $$43^{43}\equiv9^{43}\ (mod\ 17)$$ and that I...
  35. K

    Verify and Explain Binomial R.V. Identities

    If X and Y are binomial random variables with respective parameters (n,p) and (n,1-p), verify and explain the following identities: a.) P{X<=i}= P{Y>=n-i}; b.) P{X=k}= P{Y=n-k} Relevant Equations: P{X=i}=nCi *p^(i) *(1-p)^(n-i), where nCi is the combination of "i" picks given "n"...
  36. N

    Complex Analysis - Manipulating trig identities

    Homework Statement Suppose c and (1 + ic)^{5} are real, (c ≠ 0) Show that either c = ± tan 36 or c = ± tan 72The Attempt at a Solution So I considered the polar form \left( {{\rm e}^{i\theta}} \right) ^{5} and that \theta=\arctan \left( c \right) , so c = tan θ Using binomial expansion, I...
  37. K

    Confused on what should be negative when finding with half angle identities

    Homework Statement The question is to find sin 2x, cos 2x, tan 2x from the given information: sin x = -\frac{3}{5}, x in Quadrant III Homework Equations Half Angle Identities cos2x = cos^{2}x - sin^{2}x sin2x = 2sinxcosx tan2x = \frac{2tanx}{2-tan^{2}x} The Attempt at a...
  38. M

    Use standard identities to express

    Homework Statement Use standard identities to express sin(x+pi/3) in terms of sin x and cos x Homework Equations sin(a+b)=sinAcosB+sinBcosA The Attempt at a Solution sin(x)cos(pi/3)+sin(pi/3)cos(x) 0.5sinx + 0.8660cosx I'm just not sure if i need to simplify it even further...
  39. G

    Equivalence of Born and eikonal identities

    I am required to show that (i)in the upper limit of very high energies, the Born and eikonal identities are identical. (ii)that the eikonal amplitude satisfies the optical theorem. Regarding (i) I think it will involve changing from an exponential to a trig(Euler's theorem) but I could be...
  40. L

    Proving vector calculus identities using summation notation

    Homework Statement \frac{∂x_{i}}{∂x_{j}} = δ_{ij} Homework Equations \vec{r} = x_{i}e_{i} The Attempt at a Solution \frac{∂x_{i}}{∂x_{j}} = 1 iff i=j δ_{ij} = 1 iff i=j therefore \frac{∂x_{i}}{∂x_{j}} = δ_{ij} Homework Statement r^{2} = x_{k}x_{k} Homework...
  41. I

    Few Trigonometric Functions that I can’t solve involving identities? helpp

    1. Sin^2(x) = 3 – x Answer: 2.97 Attempts: 1-cos^2(x) = 3 – x cos^2(x) - x + 2 = 0 Factored it and got x = pi = 3.14 It’s a multiple choice question, and other answers were 3.02,3.09 which are few decimal places off so the answer must not be pi since it's not even a choice. Is the...
  42. P

    How can I simplify [1-(k(sin^2) θ)] using trigonometric identities?

    hi.. i came through a problem in which the expression [1-(k(sin^2) θ)] has to be simplified.. can someone help me to solve it.??
  43. S

    Proving trig identities with euler's

    Homework Statement Use Euler's identity to prove that cos(u)cos(v)=(1/2)[cos(u-v)+cos(u+v)] and sin(u)cos(v)=(1/2)[sin(u+v)+sin(u-v)] Homework Equations eui=cos(u) + isin(u) e-ui=cos(u)-isin(u) The Attempt at a Solution I was able to this with other trig identities with no...
  44. U

    Identities sin, cos, tan etc. stuff

    Homework Statement ((cos x)/(1+sin x))+((1+sin x)/(cos x)) Homework EquationsThe Attempt at a Solution multiplied the left equation by (cos x)/(cos x) and the right fraction by (1+sin x)/(1+sin x) get ((cosx)^2 +(1+sinx)^2) / (1+sinx) (cos x) and I have no idea where to go from here
  45. T

    General relativity. Bianchi identities

    Homework Statement I have a problem. I need to prove that the divergence of Einstein tensor is 0 using the bianchi identities. I have looked to several sources and I have derived an answer, but I don't fully understand some steps. Homework Equations I have uploaded a document which shows a...
  46. B

    Trig Identities: Solving for (3/5)cos2x + (3/5)sin2x

    Homework Statement (3/5)cos2x + (3/5)sin2x The Attempt at a Solution I would think the answer would be 6/5, but it looks like the book is saying 3/5. I had a similar problem to this the other day and I tried finding it in my history but I couldn't.
  47. G

    Problem with using Power-Reducing Trigonometric identities

    Homework Statement Tan^3Theta Homework Equations Tan^2Theta=1-cos2Theta/1+cosTheta The Attempt at a Solution Attached
  48. P

    Proof of No Right Identity for Operation with Two Left Identities

    If an operation has two left identities, show that it has no right identity. _{} pf/ Let e_{1} and e_{2} be left identities such that e_{1}≠e_{2}. Assume there exist a right identity and call it r. Then we have that e_{1}x=x e_{2}x=x and xr=x. From here I want to...
  49. A

    Verifying Trig. Identities

    Any/All help is appreciated :) Thanks! Homework Statement All that has to be done is proving that these two sides are equal. Basically, you just work through the problem until both sides are the same. (csc(x)-sec(x))/(csc(x)+sec(x)) = (tan(x)-1)/(tan(x)+1) Homework Equations...
  50. D

    Vector calculus identities navigation

    Homework Statement I'm reading in a fluid dynamics book and in it the author shortens an equation using identities my rusty vector calculus brain cannot reproduce. Homework Equations \vec{e} \cdot \frac{\partial}{\partial t}(\rho \vec{u}) = -\nabla\cdot (\rho\vec{u})\cdot\vec{e} -...
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