Proof of Cotangent Formula cot (α + β)

We will learn step-by-step the proof of cotangent formula cot (α + β).

Prove that, cot (α + β) = \(\frac{cot  α  cot  β   -   1}{cot  β   -   cot  α}\).

Proof: cot (α + β) = \(\frac{cos  (α  +  β)}{sin  (α  +  β)}\)

                         = \(\frac{cos  α  cos  β  -  sin  α  sin  β}{sin  α  cos  β  +  cos  α  sin  β}\)

                         = \(\frac{\frac{cos  α  cos  β}{sin  α  sin  β}  -  \frac{sin  α  sin  β}{sin  α  sin  β}}{\frac{sin  α  cos  β}{sin  α  sin  β}  +  \frac{cos  α  sin  β}{sin  α  sin  β}}\), [dividing numerator and denominator by sin α sin β].

                         = \(\frac{cot  α  cot  β  -  1}{cot  β  -  cot  α}\).            Proved

Therefore, cot (α + β) = \(\frac{cot  α  cot  β  -  1}{cot  β  -  cot  α}\).

Solved examples using the proof of cotangent formula cot (α + β):

1. Prove the identities: cot x cot 2x - cot 2x cot 3x - cot 3x cot x = 1

Solution:

We know that 3x = 2x + x

Therefore, cot 3x = cot (x + 2x)

cot 3x = \(\frac{cot  x  cot  2x  -  1}{cot  2x  +  cot  x}\)

⇒ cot x cot 2x - 1 = cot 2x cot 3x + cot 3x cot x

⇒ cot x cot 2x - cot 2x cot 3x - cot 3x cot x = 1            Proved

 

2. If α + β = 225° show that \(\frac{cot  α}{(1  +  cot  α)}\) ∙ \(\frac{cot  β}{(1  +  cot  β)}\) = 1/2

Solution:

Given, α + β = 225°

         α + β = 180° + 45°                        

 cot (α + β) = cot (180° + 45°), [taking cot on both the sides]

⇒ \(\frac{cot  α  cot  β  -  1}{cot  α  +  cot  β}\) = cot 45°

⇒ \(\frac{cot  α  cot  β  -  1}{cot  α  +  cot  β}\) = 1, [since we know cot 45° = 1]

⇒ cot α cot β - 1 = cot α + cot β 

⇒ cot α cot β = 1 + cot α + cot β

⇒ 2 cot α cot β = 1 + cot α + cot β + cot α cot β, [adding cot α cot β on both sides]

⇒ 2 cot α cot β = (1 + cot α) + cot β (1 + cot α)

⇒ 2 cot α cot β = (1 + cot α) + cot β (1 + cot α)

⇒ 2 cot α cot β = (1 + cot α)(1 + cot β)

⇒ \(\frac{cot  α}{(1  +  cot  α)}\) ∙ \(\frac{cot  β}{(1  +  cot  β)}\) = 1/2            Proved

 Compound Angle






11 and 12 Grade Math

From Proof of Cotangent Formula cot (α + β) to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Share this page: What’s this?

Recent Articles

  1. 3-digit Numbers on an Abacus | Learning Three Digit Numbers | Math

    Oct 08, 24 10:53 AM

    3-Digit Numbers on an Abacus
    We already know about hundreds, tens and ones. Now let us learn how to represent 3-digit numbers on an abacus. We know, an abacus is a tool or a toy for counting. An abacus which has three rods.

    Read More

  2. Names of Three Digit Numbers | Place Value |2- Digit Numbers|Worksheet

    Oct 07, 24 04:07 PM

    How to write the names of three digit numbers? (i) The name of one-digit numbers are according to the names of the digits 1 (one), 2 (two), 3 (three), 4 (four), 5 (five), 6 (six), 7 (seven)

    Read More

  3. Worksheets on Number Names | Printable Math Worksheets for Kids

    Oct 07, 24 03:29 PM

    Traceable math worksheets on number names for kids in words from one to ten will be very helpful so that kids can practice the easy way to read each numbers in words.

    Read More

  4. The Number 100 | One Hundred | The Smallest 3 Digit Number | Math

    Oct 07, 24 03:13 PM

    The Number 100
    The greatest 1-digit number is 9 The greatest 2-digit number is 99 The smallest 1-digit number is 0 The smallest 2-digit number is 10 If we add 1 to the greatest number, we get the smallest number of…

    Read More

  5. Missing Numbers Worksheet | Missing Numerals |Free Worksheets for Kids

    Oct 07, 24 12:01 PM

    Missing numbers
    Math practice on missing numbers worksheet will help the kids to know the numbers serially. Kids find difficult to memorize the numbers from 1 to 100 in the age of primary, we can understand the menta

    Read More