Prove that tan20° tan40° tan60° tan80° = 3

FORMULA: -

(i)

(ii) tan60° = √3,

cos90° = 0, and

(iii) 2sinAsinB = cos(A – B) – cos(A + B)

(iv) 2cosAsinB = Sin(A + B) – Sin(A – B)

tan20° tan40° tan60° tan80°

⇒ tan20° tan40° tan60° tan80° - - -

⇒ tan20° tan40° tan60° tan80°

⇒ tan20° tan40° tan60° tan80°

using formula(ii)

⇒ tan20° tan40° tan60° tan80°

multiplying with 2

⇒ tan20° tan40° tan60° tan80°

using formula(iv)

⇒ tan20° tan40° tan60° tan80°

⇒ tan20° tan40° tan60° tan80° -

using formula(iii)

⇒ tan20° tan40° tan60° tan80°

tan 20° tan 40° tan 60° tan 80°

⇒ tan20° tan40° tan60° tan80°

⇒ tan20° tan40° tan60° tan80°

⇒ tan20° tan40° tan60° tan80°

⇒ tan20° tan40° tan60° tan80° = 3