Question

Prove the following

 sin (50° − θ) − cos (40° − θ) + tan 1° tan 10° tan 20° tan 70° tan 80° tan 89° = 1

Answer 1

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