What is Trigonometric identity: Definition and 85 Discussions

In mathematics, 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. E

    B Trigonometric Identity involving sin()+cos()

    I'm trying to use the following trigonometric identity: $$ a \cos ( \omega t ) + b \sin ( \omega t ) = \sqrt{a^2+b^2} \cos ( \omega t - \phi )$$ Where ##\phi = \tan^{-1} \left( \frac{b}{a} \right)## for the following equation: $$ x(t) = -\frac{g}{ \omega^2} \cos ( \omega t) + \frac{v_o}{...
  2. brotherbobby

    To prove a trigonometric identity with tan() and cot()

    Attempt : I could not progress far, but the following is what I could do. $$\begin{align*} \mathbf{\text{LHS}} & = (\tan A+\tan B+\tan C)(\cot A+\cot B+\cot C) \\ & = 3+\tan A \cot B+\tan B \cot A+\tan A \cot C+\tan C \cot A+\tan B \cot C+\tan C \cot B\\ & = 3+\frac{\tan^2A+\tan^2B}{\tan A \tan...
  3. anemone

    MHB Prove Trig Identity: $\sin^7 x=\dfrac{35\sin x-21\sin 3x+7\sin 5x-\sin 7x}{64}$

    Prove that $\sin^7 x=\dfrac{35\sin x-21\sin 3x+7\sin 5x-\sin 7x}{64}$.
  4. L

    Evaluate this trigonometric identity

    (Sinx-2cosx)/ (cotx - sinx) Substitute tan instead of cot (Tanx(sinx-2cosx)/(1-sinx) What do I do from here I don't think what I did there is correct That's why I didn't expand the tan to sin/cos
  5. anemone

    MHB Unsolved Challenge: Trigonometric Identity

    Prove $\tan 3x=\tan \left(\dfrac{\pi}{3}-x\right) \tan x \tan \left(\dfrac{\pi}{3}+x\right)$ geometrically.
  6. PainterGuy

    B Converting the final result of a trigonometric identity back into its original form

    Hi, K₁cos(θt+φ)=K₁cos(θt)cos(φ)-K₁sin(θt)sin(φ)=K₁K₂cos(θt)-K₁K₃sin(θt) Let's assume φ=30° , K₁=5 5cos(θt+30°) = 5cos(θt)cos(30°)-5sin(θt)sin(30°) = (5)0.866cos(θt)-(5)0.5sin(θt) = 4.33cos(θt)-2.5sin(θt) If only the final result, 4.33cos(θt)-2.5sin(θt), is given, how do I find the original...
  7. Krushnaraj Pandya

    Proof of an inverse trigonometric identity

    Homework Statement Show that ##\arcsin 2x \sqrt{1-x^2} = 2 \arccos{x}## when 1/√2 < x < 1 Homework Equations All trigonometric and inverse trigonometric identities, special usage of double angle identities here The Attempt at a Solution I can get the answer by puting x=cosy, the term inside...
  8. R

    MHB Stuck on a trigonometric identity proof....

    $\frac{1 -\cos A}{1 + \cos A} = (\cot A - \csc A)^2$
  9. DeathbyGreen

    I Infinite series of trigonometric terms

    I'm trying to make an approximation to a series I'm generating; the series is constructed as follows: Term 1: \left[\frac{cos(x/2)}{cos(y/2)}\right] Term 2: \left[\frac{cos(x/2)}{cos(y/2)}-\frac{sin(x/2)}{sin(y/2)}\right] I'm not sure yet if the series repeats itself or forms a pattern...
  10. S

    MHB Trigonometric Identity

    Prove $Cos^6A+Sin^6A=1-3 \hspace{0.2cm}Sin^2 A\hspace{0.02cm}Cos^2A$ So far, $Cos^6A+Sin^6A=1-3 \hspace{0.2cm}Sin^2 A\hspace{0.02cm}Cos^2A$ $L.H.S=(Cos^2A)^3+(Sin^2A)^3$ $=(Cos^2A+Sin^2A)(Cos^4A-Cos^2ASin^2A+Sin^4A)$...
  11. D

    Solve acos²θ+bsinθ+c=0 for all values 0≤θ≤360°

    Homework Statement Solve acos²θ+bsinθ+c=0 for all values 0≤θ≤360° a=16 b=6 c=-12 So 16cos²θ+6sinθ-12=0 Homework Equations Cos²x=1-Sin²x The Attempt at a Solution Identity: Cos²x=1-Sin²x 16(1-Sin²θ)+6Sinθ-12=0 16-16Sin²θ+6Sinθ-12=0 6Sinθ-16Sin²θ=12-16=-4 Divide by 2(?) 3Sinθ-8Sin²θ=-2...
  12. FritoTaco

    Verify the Trigonometric Identity

    Homework Statement Problem 1: csc(tan^{-1}\dfrac{x}{2})=\sqrt{\dfrac{x^{2}+4}{x}} Problem 2: \sqrt{\dfrac{1-sinx}{1+sinx}}=\dfrac{|cosx|}{1+sinx} Homework Equations Quotient Identities tan\theta=\dfrac{sin\theta}{cos\theta} cos\theta=\dfrac{cos\theta}{sin\theta} Pythagorean Identites...
  13. 8

    MHB Trigonometric Identity: Tan^2-Sin^2 = Sin^2 Cos^2

    \tan\left({^2}\right)-\sin\left({^2}\right)=\tan\left({^2}\right) \sin\left({^2}\right) i keep on getting \sin\left({^2}\right)-\sin\left({^2}\right) \cos\left({^2}\right)=\sin\left({^2}\right) \sin\left({^2}\right) \cos\left({^2}\right)...
  14. 8

    MHB Home work help: proving a trigonometric identity

    1 ___________ =csc2\theta-csc\thetacot\theta 1+cos\theta
  15. E

    I Is this a Trigonometric Identity?

    I have encountered this equation: ##\cos^2 \gamma = \cos^2 \alpha \cdot \cos^2 \beta## According to the paper, this is a trigonometric identity, but this is the first time I have encountered this. The angles ##\alpha## and ##\beta## are somewhat similar to the components of the distance...
  16. Draconifors

    Quick Trigonometric Identity Question

    Hi! I have an integral to solve (that's not the point, though) and the inside of the integral is almost a trig identity: 1. Homework Statement ##sin\frac{(x+y)} {2}*cos\frac{(x-y)} {2} ## Homework Equations I noticed this was very similar to ##sinx+siny = 2sin \frac{(x+y)} {2} *...
  17. J

    Confused about proof of "sin(θ + Φ) = cosθsinΦ + sinθcosΦ"

    Hi, This is also a sort of geometry question. My textbook gives a proof of the relation: sin(θ + Φ) = cosθsinΦ + sinθcosΦ. It uses a diagram to do so: http://imgur.com/gLnE2Fn sin (θ + Φ) = PQ/(OP) = (PT + RS)/(OP) = PT/(OP) + RS/(OP) = PT/(PR) * PR/(OP) + RS/(OR) * OR/(OP) = cosθsinΦ +...
  18. Rectifier

    Proving Trigonometric Identity: tan(x/2) = (1-cos(x))/sin(x)

    The problem Show that the left side is equal to right side ## tan (\frac{x}{2}) = \frac{1-cos(x)}{sin(x)} ## The attempt ##\tan(\frac{x}{2}) = \frac{ sin(\frac{x}{2}) }{ cos (\frac{x}{2}) } = \frac{ sin^2(\frac{x}{2}) }{ cos ^2 (\frac{x}{2}) } = \frac{\frac{1-cos(x)}{2}}{\frac{1+cos(x)}{2}} =...
  19. B

    Sin^4Ө =3/8-3/8cos(2Ө) Prove the following trigonometric identity

    Homework Statement Prove the following trigonometric identity. The question is sin^4Ө =3/8-3/8cos(2Ө) Homework Equations I think I'm supposed to use the power reducing formulas for trigonometric identities which are sin^2(u)= (1- cos(2u))/2 cos^2(u)=(1+cos(2u))/2 *Let u represent any...
  20. N

    Proof using hyperbolic trig functions and complex variables

    1. Given, x + yi = tan^-1 ((exp(a + bi)). Prove that tan(2x) = -cos(b) / sinh(a)Homework Equations I have derived. tan(x + yi) = i*tan(x)*tanh(y) / 1 - i*tan(x)*tanh(y) tan(2x) = 2tanx / 1 - tan^2 (x) Exp(a+bi) = exp(a) *(cos(b) + i*sin(b))[/B]3. My attempt: By...
  21. M

    Master Trigonometric Identities with Double Angle Techniques

    I am doig trigonometric identities and i got this one, (all will be in the picture the solution and my work) i used the double angle for this but i am afraid i didn't get the exact idea, just guessing, good guessing, so i want to know how is the proper way to reach the solution
  22. kostoglotov

    Different answers: integral table vs trig identity solutions

    EDIT: I figured out my mistake...no option to delete silly post. Oh well. 1. Homework Statement The problem is: use iterated integrals in polar form to find the area of one leaf of the rose-shaped curve r = cos(3*theta). My setup agrees exactly with the solutions manual...but then something...
  23. E

    Proving a Trigonometric Identity

    Homework Statement Prove that: \cos^6{(x)} + \sin^6{(x)} = \frac{5}{8} + \frac{3}{8} \cos{(4x)} Homework Equations I am not sure. I used factoring a sum of cubes. The Attempt at a Solution I tried \cos^6{(x)} + \sin^6{(x)} = \cos^4{(x)} - \cos^2{(x)} \sin^2{(x)} + \sin^4{(x)} . But I...
  24. E

    MHB Proving a trigonometric identity II

    Hi, I need help proving the following trig identity, (2sinx)\overline{secxtan(2x)}=2cos^2x-csc^2x+cot^2x I have tried starting from the left hand side, the right hand side, and doing both together, but nothing seems to work. One of the ways I tried: LHS...
  25. E

    MHB Proving a trigonometric identity

    Hi, I need help proving the following trig identity: \frac{\cot^2(x)-\cot(x)+1}{1-2\tan(x)+\tan^2(x)}=\frac{1+\cot^2(x)}{1+\tan^2(x)} Me and my friend have spent several hours determined to figure this out, starting from the left hand side, the right hand side, and doing both together, but...
  26. M

    MHB Proving a trigonometric identity

    How prove $\cos\frac{8\pi}{35}+\cos\frac{12\pi}{35}+\cos\frac{18\pi}{35}=\frac{1}{2}\cdot\left(\cos\frac{\pi}{5}+\sqrt7\cdot\sin\frac{\pi}{5}\right)$?
  27. Ganesh Ujwal

    How do I prove this seemingly simple trigonometric identity

    Mod note: Fixed the LaTeX. ##a=sinθ+sinϕ## ##b=tanθ+tanϕ## ##c=secθ+secϕ##Show that, ##8bc=a[4b^2 + (b^2-c^2)^2]## I tried to solve this for hours and have gotten no-where. Here's what I've got so far : ##a= 2\sin(\frac{\theta+\phi}{2})\cos(\frac{\theta-\phi}{2}) ## ## b =...
  28. C

    Trigonometric identity double definite integral

    Double integral of (52-x^2-y^2)^.5 2<_ x <_ 4 2<_ y <_ 6 I get up to this simplicity that results in a zero! 1-cos^2(@) - sin^2(@) = 0 This identity seems to be useless. HELP PLEASE.
  29. S

    MHB Can the derivative of the given integral be simplified to -A?

    Can it be proved? \left(\frac{-2\sin A}{1-\cos A}\right)\cos\left(\frac{A}{2}\right)\tan^{-1}\left[\cos \left(\frac{A}{2}\right)\right]=\frac{\pi^2-4A^2}{8}
  30. MarkFL

    MHB Verify Trig Identity: 1+cosx+cos2x=1/2+(sin5/2x)/(2sin1/2x) - Catlover0330

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  31. S

    MHB Trigonometric Identity Questions

    Your help will be greatly appreciated! Thanks!1. The expression \(\sin\pi\) is equal to \(0\), while the expression $\frac{1}{\csc\pi}$ is undefined. Why is $\sin\theta=\frac{1}{\csc\theta}$ still an identity? 2. Prove $\cos(\theta + \frac{\pi}{2})= -\sin\theta$
  32. MarkFL

    MHB Hello's question at Yahoo Answers regarding proving a trigonometric identity

    Here is the question: I have posted a link there to this topic so the OP can see my work,
  33. hxthanh

    MHB Summation: trigonometric identity

    Prove that: $\displaystyle\sum_{k=0}^n \frac{\cos(k x)}{\cos^kx} = \frac{1+(-1)^n}{2\cos^nx} + \dfrac{2\sin\big(\lfloor\frac{n+1}{2}\rfloor x\big) \cos\big(\lfloor\frac{n+2}{2}\rfloor x\big)} {\sin x\cos^n x} \qquad\qquad (\frac{2x}{\pi}\not\in \mathbb Z)$ *note: $\lfloor x\rfloor$ is floor...
  34. anemone

    MHB Trigonometric Identity

    Prove \frac{\sin\left(5\tfrac{3}{4}^{\circ} \right)}{\cos\left(17\tfrac{1}{4}^{\circ} \right)}+\frac{\sin\left(17\tfrac{1}{4}^{\circ} \right)}{\cos\left(51\tfrac{3}{4}^{\circ} \right)}+\frac{\sin\left(51\tfrac{3}{4}^{\circ} \right)}{\cos\left(155\tfrac{1}{4}^{\circ}...
  35. A

    Proving This Trigonometric Identity

    1. Prove:\frac{cscx +cotx}{cscx-cotx} = \frac{1+2cosx+cos^2x}{sin^2x} Homework Equations tanx = \frac{sinx}{cosx} cotx = \frac{cosx}{sinx} cscx secx cotx sin^2x + cos^2x = 1 The Attempt at a Solution Left side: =\frac{cscx +cotx}{cscx-cotx} =\frac{1/sinx + cosx / sinx}{1/sinx - cosx/sinx}...
  36. E

    Trigonometric identity from Euler's intro to analysis of infinite

    So I'm trying to get through euler's introduction to the analysis of the infinite so I could eventually read his books on calculus but I'm stuck somewhere and can't seem to figure out how he equates this identity so by expanding I get sin(2y) * cos(z) + cos(2y) * sin(z). I get that the...
  37. C

    Sum to Product Trigonometric identity does not work

    "Sum to Product" Trigonometric identity does not work Hi, The identity sin(u) + sin(v) = 2 * sin (\frac{u+v}{2}) * cos(\frac{u-v}{2}) http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identities Does not always work. I put the equation : (sin(u)...
  38. MarkFL

    MHB Katlynsbirds' question at Yahoo Answers regarding inverse trigonometric identity

    Here is the question: Here is a link to the question: Prove the identity, pre calc!? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  39. S

    Trigonometric Identity Proof: v cosδ = V(1-cosβ) + u cos(α-β)

    Homework Statement Given the following two triangles: Show that v \cos{\delta} = V(1-\cos{\beta})+u\cos(\alpha - \beta) The Attempt at a Solution Using the cosine law I've got: v^{2}=x^{2}+V^{2}-2xV\cos{(\theta + \beta)} and u^{2}=x^{2}+V^{2}-2xV\cos{(\theta)} I figured maybe using the...
  40. P

    Prove the trigonometric identity

    Homework Statement (question attached) Homework Equations The Attempt at a Solution Checking solution.. pretty sure I did this wrong. (solution attached)
  41. MarkFL

    MHB Mangoqueen54's question at Yahoo Answers involving a trigonometric identity

    Here is the question: Here is a link to the question: http://answers.yahoo.com/question/index?qid=20130130130636AAOqgvz I have posted a link there so the OP can find my response.
  42. K

    Trigonometric Identity Problem

    Homework Statement Prove the Identity sinθ/(1+cosθ) = 1-cos(θ)/sinθ Homework Equations sinθ/cosθ = tanθ sin^2θ + cos^2θ = 1 The Attempt at a Solution sinθ/(1 + cosθ) = LS cosθtanθ/(1+cosθ) = LS cosθtanθ/(sin^2θ + cos^2θ + cosθ) = LS cosθtanθ/(tan^2θcos^2θ +...
  43. Feodalherren

    Trigonometric Identity Homework: Solving with Sin and Cos Formulas

    Homework Statement Homework Equations Any trig formulas The Attempt at a Solution The yellow paper is me switching everything to sin and cos to see if that helps but it doesn't. I'm completely stuck here.
  44. J

    Proving the trigonometric identity

    Homework Statement To prove that \sum over m=1 to 15 of sin(4m-2) = 1/4sin2, where all angles are in degress Homework Equations The Attempt at a Solution Tried to solve it using identity sinx+siny=2sin((x+y)/2)cos((x-y)/2)..but all attempts failed..help
  45. H

    Proving this trigonometric identity

    Homework Statement Show that : 1 + cos(2∏/5)= 2 cos(∏/5) Homework Equations cos(2x) = cos^2(x)-sin^2(x) cos^2(x)+sin^2(x) = 1 The Attempt at a Solution I have tried using the two formulas above but i couldn't show the required result.
  46. J

    Trigonometric Identity: Double Angle.

    Homework Statement Here is the question given: A blade for a lawnmower consists of two parts made of the same material and joined together as shown: The length OP is one unit in length and MPQN is square in shape. Develop an equation for the cross-sectional area of the blade and...
  47. P

    Can Trigonometric Identities Be Proven Using Different Methods?

    Homework Statement Prove that: tan^2∅/tan∅ - 1 + cot^2∅/cot∅ - 1 = 1 + sec∅cosec∅ Homework Equations The Attempt at a Solution I have solved the question taking tan∅ = sin∅/cos∅. But I want to solve it some other way.
  48. cocopops12

    How can someone prove the following trigonometric identity?

    it's bothering my brain..i thought about it many times...i can't make intuition of it can anyone prove it? oh by the way... C = Sqrt[A^2 + B^2] and theta is equal to arctan(B/A)
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