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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) = 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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11 and 12 Grade Math
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