Question

Tan70°=2Tan50°+Tan20°, prove it.

Answer 1

tan( 50° + 20°) = tan70°

use the formula,
tan(A + B) = (tanA + tanB)/(1 - tanA.tanB)

(tan50° + tan20°)/(1 - tan50°.tan20°) = tan70°
(tan50° + tan20°) = tan70°(1 - tan50°.tan20°)
tan50° + tan20° = tan(90°-20°)(1-tan50°.tan20°)
tan50° +tan20° = cot20°(1 - tan50°.tan20°)
tan50° + tan20° = cot20° - tan50°.tan20°.cot20°

[use cot20° = tan70° also cot20°.tan20°]

tan50° + tan20° = tan70° - tan50°
2tan50° + tan20° = tan70°
tan70° = 2tan50° + tan20°
hence proved

Answer 2

Hey,
Tan70°=Tan(50°+20°) =Tan50°+Tan20°/1-Tan50°*Tan20°(By using formula tan(a+b) =tana+tanb/1-tana*tanb)

Or, Tan70°(1-Tan50°*Tan20°) =Tan50°+Tan20°

Or, Tan70°-Tan70°*Tan50°*Tan20°=Tan50°+Tan20°

Or, Tan70°=Tan70°*Tan50°*Tan20°+Tan50°+Tan20°

=Tan(90°-20°) *Tan50°*Tan20°+Tan50°+Tab20°

=cot20°Tan50°*Tan20°+Tan50°+Tan20°

=Tan50°+Tan4?50°+Tan20°=2Tan50°+Tan20°.