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

    Prove trigonometric identity and determine a counterexample

    Homework Statement cos(x-y)cosy-sin(x-y)siny=cosx a.try to prove that the equation is an identity b. determine a counterexample to show that it is not an identity Homework Equations cos(x-y) = cosxcosy+sinxsiny sin(x-y) = sinxcosy-cosxsiny The Attempt at a Solution a.Left side of...
  2. cocopops12

    What is the trigonometric identity used to simplify this expression?

    Could anyone please tell me how are those two expressions equivalent?! what are the identities used?
  3. T

    Proving LS=RS in Trigonometry?

    Homework Statement http://img829.imageshack.us/img829/3413/daumequation13237287425.png Prove that LS=RS. Homework Equations There are no relevant equations. The Attempt at a Solution http://img829.imageshack.us/img829/3413/daumequation13237287425.png
  4. U

    Is there a trigonometric identity for this ?

    Hi, I am trying to figure out what the result is when adding two sinusoids of the same frequency but with different phase and amplitudes. Specifically I want to know if the result is always another sinusoid of the same frequency. For the case of the the same amplitude I have: cos(wt) +...
  5. R

    How can I use trigonometric identities to simplify this equation?

    In the following question I figure that i need to prove that h holds true with the trigonometric identity subbed into the denominator. I'm not sure how to simplify the equation any further after that though. Can someone provide any insight...
  6. B

    Help proving trigonometric identity

    Homework Statement cos\theta/1-tan\theta+sin\theta/1-cot\theta=sin\theta+cos\theta Homework Equations The Attempt at a Solution
  7. T

    Show this simple trigonometric identity add up to 0

    Homework Statement \cos (\frac{(-1)\pi x}{L})-\cos (\frac{3\pi x}{L}) Homework Equations The Attempt at a Solution the first cosine is the same as positive but is the second cosine simply equal to \cos (\frac{\pi x}{L})? thanks!
  8. P

    What is the Origin of the Trigonometric Identity?

    I'm reading through a solution to a problem and at one point the following identity is used: \frac{1-\cos(\beta)}{\sin(\beta)}=\frac{\sin(\frac{\beta}{2})}{\cos(\frac{\beta}{2})} I've been trying to figure out where this comes from but with haven't got it yet. Any ideas?
  9. I

    Verifying this Trigonometric Identity

    Hey guys. How are you all doing? I'm helping my younger brother out with his trigonometry homework. He is dealing with verifying trigonometric identities. However, he has the problem that I am getting nowhere with. Hope you all can help. Thanks in advance. :) Homework Statement Verify...
  10. K

    Trigonometric identity for inverse tan

    Hello, Could you please clarify if this is correct: If tan^(-1)(x) = Pi/2 - tan^(-1)(1/x) Then if we have (ax) as the angle where a is a constant, do we get: tan^(-1)(ax) = Pi/2 - tan^(-1)(a/x) or does the constant go on the bottom with the x? i.e. or: tan^(-1)(ax) = Pi/2 -...
  11. J

    Precalculus: proving trigonometric identity

    Homework Statement prove that: tan(1+cos(x))^2 = 1-cos(x) Homework Equations trig identities, like the pythagorean, sum/difference, double/half angle identities, power reducing identities, etc... The Attempt at a Solution i'm not sure where to start; i tried using the pythagorean...
  12. K

    Help with proving a trigonometric identity

    Hi I've got this problem which has really been bothering me. How are you supposed to prove that: (Sin[A] Sin[2 A] + Sin[3 A] Sin[6 A])/(Sin[A] Cos[2 A] + Sin[3 A] Cos[6 A]) is identicle to tan[5A]. I am almost sure that I've got to use the factor formulae, but I've had no luck. Maybe...
  13. B

    Trigonometric Identity and Differential Equation question

    I'm looking over the differential equation describing a hanging cable in a textbook, and I probably need to review my trigonometric derivatives and integrals again because I'm not seeing how they got the following: \frac{dy}{dx} = tan(\phi) \frac{ws}{T_0} \frac{d^2y}{dx^2} =...
  14. N

    A Generalized trigonometric identity for Cos(x_1++x_n)?

    Hi Everyone, Do there exist any explicit formula for Cos(x_1+x_2+...+x_n) as a sum of products of Sin(x_i) & Cos(x_i)? Or we need to expand using Cos(A+B), Sin(A+B) again & again? If it exists then what is about Sin(x_1+x_2+...+x_n)? [It is understood that there will be 2^(n-1) number of...
  15. L

    Understanding the Trigonometric Identity: cos^2 x = 1/2 + cos(2x)/2

    Why is cos^2 x = \frac{1}{2} + \frac{cos(2x)}{2} ?
  16. L

    Are the given trigonometric identities correct?

    Homework Statement tanX+tan Y / 1-tanXtanY= cotX+cotY/cotXcotY-1 Homework Equations tan= sin/cos cot= cos/sin The Attempt at a Solution
  17. L

    Can This Trigonometric Identity Be Verified?

    Homework Statement verify the identity : 1+sec(-∅)/sin(-∅+tan(-∅) = -csc ∅ Homework Equations The Attempt at a Solution 1+sec∅/-sin-tan∅ = -csc∅ I don't know where to start from, does anyone have any idea?
  18. B

    Proving Trigonometric Identity: cosx/1-sinx + 1-sinx/cosx = 2secx + 2tanx

    cosx/1-sinx + 1-sinx/cosx = 2secx + 2tanx I can get to 2secx + tanx - tanx, any help is appreciated
  19. C

    A Trigonometric Identity Probelm

    [SOLVED] A Trigonometric Identity Probelm If I have sin^2 2x would I be able to apply the identity sin^2x = (1/2)(1-cos2x) to get this: sin^2 2x = 2(1/2)(1 - cos^2 x) Similarly, if I had sin^2 2x + cos^2 2x would I be able to use the identity sin^2 x + cos^2 x = 1 to get: sin^2 2x +...
  20. T

    Arc length of a curve (trigonometric identity)

    Homework Statement find arc length of the segment of the 2space curbe that is defined by the parametric equations x(t) = t-sin(t) y(t) = 1+cos(t) 0 ≤ t ≤ 4π The Attempt at a Solution I've found dx/dt and dy/dt respectively and put them into the arc length equation, i.e...
  21. S

    Trigonometric Identity Problem

    Homework Statement http://img206.imageshack.us/img206/9099/titleol2.jpg http://g.imageshack.us/g.php?h=206&i=titleol2.jpg Show the above statement is equivalent to : sec (2x) + tan (2x) Homework Equations The Attempt at a Solution First attempt in which I used the...
  22. J

    Prove this trigonometric identity

    Homework Statement Verify the possibility of an identity graphically. (Completed this part) Then, prove each identity algebraically. \dfrac{sinx+tanx}{cosx+1}=tanx Homework Equations tan\theta=\dfrac{sin\theta}{cos\theta} cot\theta=\dfrac{cos\theta}{sin\theta}...
  23. L

    Proving Trigonometric Identities: Attempt and Solution

    Homework Statement I'm attempting to prove that 1 - sin^2 t /(1 + cos t) - cos^2/(1+tan t) = cos t sin t 2. The attempt at a solution I've tried various approaches. The most promising has the LHS reduced to: (sin t cos t (1 + cos t + sin t cos t))/((1 + cos t)(cos t + sin t))...
  24. L

    Proving a Trigonometric Identity

    Im supposed to verify that (1-sinx)/(1+sinx) = (secx-tanx)^2 RHS = (secx-tanx)^2 = (1/cosx - sinx/cosx)^2 = [(1-sinx) / cosx]^2 = [(1-sinx)(1-sinx)]/cosx^2 = (1-2sinx+sinx^2)/(1-sinx^2) From here, I'm feeling pretty confused. I'm not even sure if all my values are correct.
  25. Q

    Help with a Trigonometric identity

    Help with a Trigonometric identity... Homework Statement (sin x + sin 2x + sin 4x) / (cos x + cos 2x + cos 4x) = tan 2x Homework Equations sin 2x = 2sinxcosx; cos 2x = cos^2x - sin^2x The Attempt at a Solution solving left side, =[sin x + sin 2x + sin (2x + 2x)]/[cos x + cos...
  26. H

    Deriving a trigonometric identity

    For a homework assignment I'm supposed to prove that sin(x)^2+cos(x)^2=1, using only the following identities (along with algebraic operations): sin(-x)=-sin(x) cos(-x)=cos(x) cos(x+y)=cos(x)cos(y)-sin(x)sin(y) sin(x+y)=sin(x)cos(y)+cos(x)sin(y) I can't figure this out, because as far...
  27. K

    Transforming Trigonometric Identities: Solving 2(cosx)^2+2cosxsinx=0

    Which trigonometric identitiy are used to transform 2(cos(x))^2+2cos(x)sin(x) into 0?
  28. G

    Trigonometric identity forced oscillations

    Homework Statement http://www.jyu.fi/kastdk/olympiads/2004/Theoretical%20Question%203.pdf http://www.jyu.fi/kastdk/olympiads/2004/Solution%203.pdf Question A- (b) They use some trigomentric identity that I don't understand, which one is it? Thanks in advance. Homework...
  29. C

    Proving this trigonometric identity

    Homework Statement Prove the following identity: (1 + cos \theta)^2 + sin^2\theta = 2(1 + cos \theta) Homework Equations cos^2 \theta + sin^2 \theta = 1 The Attempt at a Solution I've squared out the first bracket so that it becomes 1 + cos^2 \theta and multiplied out the second bracket so...
  30. D

    Proving a Trigonometric Identity Using Exponential Functions

    Hi. I need to prove the following identity \arccos{z} =i \ln { z + (z^2 -1)^\frac{1}{2} } I was given a hint to write \cos{A}=z, then rewrite \cos{A} in terms of the exponential. \cos{A}=\frac{\exp{iA}+\exp{-iA}}{2}=z I took the log on both sides and got stuck at that...
  31. K

    Trigonometric Identity in My Book: Understanding (cos4x)^2 = 1+cos8x

    In my book, (cos4x)^2 is written 1+cos8x without referring to any formula. Which trig. identity is used here?
  32. K

    Trigonometric Identity Problems

    Can anyone help me solve the following problems? sec theta -1/1-cos theta = sec theta tan (pie/2 - theta) tan theta =1 Thanks
  33. C

    Trigonometric Identity for 1/2csc(THETA)sec(THETA)

    Hi, I have a question about a problem: 1/2csc(THETA)sec(THETA) I have to find the identity for it and I know that csc is 1/sin and sec is 1/cos but after that I don't see where it can lead to a simple answer. If anyone knows it would be helpful, thanks!
  34. J

    Trigonometric Identity? - Deriving the Double Angle Formulas for Sine and Cosine

    Trigonometric Identity?? I just can't figure this out. I don't think I've covered enough material to do this. Can anyone help? I've put the entire question but I am sure all i need is a little explanation and maybe the first answer and i could do the rest. :confused: Let z =...
  35. E

    Help me derive Lagrange's Trigonometric Identity

    Using the fact that z=e^ibeta and the identity 1 + z + z^2...+z^n=1-z^(n+1)/1-z Help me derive 1 + cosbeta +cos2beta + ... +cosnbeta= 1/2 + sin((2n+1)beta/2)/2sin(beta/2) where beta is between 0 and 2pi
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