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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) = cotαcotβ1cotβcotα.

Proof: cot (α + β) = cos(α+β)sin(α+β)

                         = cosαcosβsinαsinβsinαcosβ+cosαsinβ

                         = cosαcosβsinαsinβsinαsinβsinαsinβsinαcosβsinαsinβ+cosαsinβsinαsinβ, [dividing numerator and denominator by sin α sin β].

                         = cotαcotβ1cotβcotα.            Proved

Therefore, cot (α + β) = cotαcotβ1cotβcotα.

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 = cotxcot2x1cot2x+cotx

⇒ 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 cotα(1+cotα)cotβ(1+cotβ) = 1/2

Solution:

Given, α + β = 225°

         α + β = 180° + 45°                        

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

cotαcotβ1cotα+cotβ = cot 45°

cotαcotβ1cotα+cotβ = 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)

cotα(1+cotα)cotβ(1+cotβ) = 1/2            Proved

 Compound Angle






11 and 12 Grade Math

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