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

• Proof of Compound Angle Formula sin (α + β)
• Proof of Compound Angle Formula sin (α - β)
• Proof of Compound Angle Formula cos (α + β)
• Proof of Compound Angle Formula cos (α - β)
• Proof of Compound Angle Formula sin 22 α - sin 22 β
• Proof of Compound Angle Formula cos 22 α - sin 22 β
• Proof of Tangent Formula tan (α + β)
• Proof of Tangent Formula tan (α - β)
• Proof of Cotangent Formula cot (α + β)
• Proof of Cotangent Formula cot (α - β)
• Expansion of sin (A + B + C)
• Expansion of sin (A - B + C)
• Expansion of cos (A + B + C)
• Expansion of tan (A + B + C)
• Compound Angle Formulae
• Problems using Compound Angle Formulae
• Problems on Compound Angles

11 and 12 Grade Math

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