# 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) = cot α cot β + 1/cot β - cot α.

Proof: cot (α - β) = cos(α - β)/sin (α - β)

= cos α cos β + sin α sin β/sin α cos β - cos α sin β

= cos α cos β/sin α sin β + sin α sin β/sin α sin β/sin α cos β/sin α sin β - cos α sin β/sin α sin β, [dividing numerator and denominator by sin α sin β].

= cot α cot β + 1/cot β - cot α                     Proved

Therefore, cot (α - β) = cot α cot β + 1/cot β - cot α.

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

1. Find the value of cot 15°.

Solution:

cot 15°

= cot (45° - 30°)

= cot 45° cot 30° + 1/cot 30° - cot 45°

= 1 ∙ √3 + 1/√3 - 1

= √3 + 1/√3 - 1

= (√3 + 1)^2/(√3 - 1) (√3 + 1)

= 3 + 2√3 + 1/3 – 1

= 4 + 2√3/2

= 2 + √3

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