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Question
Prove the following
sin (50° − θ) − cos (40° − θ) + tan 1° tan 10° tan 20° tan 70° tan 80° tan 89° = 1
Solution
Sin (50 + θ) = cos (90 – (50 + θ)) = cos (40 – θ)
Tan 1 = tan (90° − 89°) ∙ cot 89°
Tan 10° = tan (90° - 80°) = cot 80°
Tan 20° = tan (90° - 70°) = cot 70°
⇒ cos (40° - θ) – cos (40 - θ) = cot 89° tan 89° . cot 80° . cot 70° tan 70°
Cot . tan θ = 1
= 1 ∙ 1 ∙ 1 = 1
LHS = RHS
Hence proved
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