Question

find the value tan20 + 2 tan50 =?
(a) tan70 (b)cot70 (c)sin 70 (d) tan30
(all angles are in degree)
With full solution plz

Answer 1

Solution:

Tan 20 + 2 Tan 50 = Tan 70

Proof:

RHS:

Tan 70 = Tan ( 50 + 20 )

$Tan ( A + B ) = \frac{ Tan \ A \ + \ Tan \ B }{ 1 - \ Tan A . Tan \ B }$

=> $Tan ( 50 + 20 ) = \frac{Tan 50 + Tan 20}{ 1 - Tan 50. Tan 20 }$

Cross multiplying we get,

=> $Tan 70 \  ( 1 - Tan 50 \ . \ Tan 20 ) = Tan\ 50 + Tan\ 20 \\\\  => Tan\ 70 - Tan\ 70\ . \ Tan\ 50 \ . \ Tan \ 20  = Tan\ 50 \ + \ Tan\ 20$

$We \ know \ that, \ Tan\ 70\ .\ Tan\ 20 = Tan \ 70\ .\ Cot\ 70 = 1$

Hence we get,

=> $Tan\ 70 - (1\ .\ Tan\ 50) = Tan\ 20\ +\ Tan\ 50 \\\\ => Tan\ 70 = Tan\ 20 + 2\ Tan\ 50$

$\boxed{ Hence Proved}$

Hope my answer helped !