Turn on thread page Beta
    • Thread Starter
    Offline

    12
    ReputationRep:
    Can someone please prove this identity:

    cot x − cot 2x ≡ cosec 2x



    Thanks in advance.
    Offline

    16
    ReputationRep:
    The key is to remember the identity,
    sin 2x = 2 cos x sin x
    and thus,

    sin x = sin 2x / 2 cos x (1)

    and

    cos 2x = 2cos^2 x - 1 (2)

    and remember to work in cosines and sines.
    Ok, so we start with (LHS)

    cot x - cot 2x

    (cos x/sin x) - (cos 2x/sin 2x)

    (cos x/(sin 2x / 2 cos x)) - (cos 2x/sin 2x) (using 1)

    (2 cos^2/sin 2x) - (cos 2x/sin 2x)

    (2 cos^2/sin 2x) - (2cos^2 x - 1/sin 2x) (using 2)

    1 / sin 2x

    Which is cosec 2x. LHS=RHS. Just a few lines
    Offline

    0
    ReputationRep:
    (Original post by aiman)
    cot x − cot 2x ≡ cosec 2x
    cot x − cot 2x ≡ cosec 2x
    1/tanx − 1/tan[2x]
    => cosx/sinx − cos[2x]/sin[2x]
    = (cosxsin[2x] − cos[2x]sinx) / sin[2x]sinx
    = (2cos²[x]sin[x] − (1-2sin²x)sinx) / sin[2x]sinx
    = (2cos²[x]sin[x] − sinx + 2sin³x)) / sin[2x]sinx
    = (2(1-sin²[x])sin[x] − sinx + 2sin³x)) / sin[2x]sinx
    = (2sinx - 2sin³[x] − sinx + 2sin³x) / sin[2x]sinx
    = (2 - 2sin²[x] − 1 + 2sin²x) / sin[2x]
    => 1 / sin[2x] => cosec[2x]

    ∴ cot x − cot 2x ≡ cosec[2x]
    Offline

    16
    ReputationRep:
    Mine was better :P lol
    • Thread Starter
    Offline

    12
    ReputationRep:
    Thanks alot both of you were wonderful!
    Offline

    0
    ReputationRep:
    cot 2x + cosec 2x = cot x

    how would one go about proving this?


    thanks
    • PS Helper
    Offline

    14
    PS Helper
    (Original post by Sajee)
    cot 2x + cosec 2x = cot x

    how would one go about proving this?


    thanks
    This is exactly the same as the problem in the first post of this (six year old) thread, just rearranged; a complete worked answer is provided a few posts up.
    • Community Assistant
    Offline

    20
    ReputationRep:
    Community Assistant
    (Original post by nuodai)
    This is exactly the same as the problem in the first post of this (six year old) thread, just rearranged; a complete worked answer is provided a few posts up.
    Slightly worrying that the thread dredger didn't notice this.
    Offline

    0
    ReputationRep:
    (Original post by nuodai)
    This is exactly the same as the problem in the first post of this (six year old) thread, just rearranged; a complete worked answer is provided a few posts up.
    thank you,
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: December 6, 2011

University open days

  • Heriot-Watt University
    School of Textiles and Design Undergraduate
    Fri, 16 Nov '18
  • University of Roehampton
    All departments Undergraduate
    Sat, 17 Nov '18
  • Edge Hill University
    Faculty of Health and Social Care Undergraduate
    Sat, 17 Nov '18
Poll
Black Friday: Yay or Nay?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Equations

Best calculators for A level Maths

Tips on which model to get

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.