Pembuktian TrigLog (Trigonometri dan Logaritma)

(i) cot (⁹/₈ π) = 1 + √2

Ruas kiri = cot (⁹/₈ π)
= cot ½(⁹/4 π)
= cot ½(⁹/4 π – π)
= cot ½(¼ π)

• tan ½α = sin α / (1 + cos α)
  ⇒ cot ½α = (1 + cos α) / sin α

= (1 + cos (¼ π)) / sin (¼ π)
= (1 + ½√2) / (½√2)
= 2 · [(1 + ½√2) / √2]
= √2(1 + ½√2)
= √2 + ½·2
= √2 + 1
= 1 + √2 = ruas kanan
⇒ terbukti
_____________

(ii) 3 – 2√2 = tan² (⅛ π)

Ruas kanan = tan² (⅛ π)
= tan² ½(¼ π)

• tan ½α = √[(1 – cos α) / (1 + cos α)]
  ⇒ tan² ½α = (1 – cos α) / (1 + cos α)

= (1 – cos (¼ π)) / (1 + cos (¼ π))
= (1 – ½√2) / (1 + ½√2)
= (1 – ½√2)² / (1 + ½√2)(1 – ½√2)
= (1 – √2 + ½) / (1 – ½)
= (1 ½ – √2) / ½
= 2(1 ½ – √2)
= 3 – 2√2 = ruas kiri
⇒ terbukti
_____________

(iii) 1 + 1 = log 100

Ruas kanan = log 100
= log (10×10)
= log 10 + log 10
= 1 + 1 = ruas kiri
⇒ terbukti