Using tan(a+b)=tanA+tanB/1-tanA×tanB. Prove that tan70°=tan20°+2tan50° ?

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Answer 1 · 2018-06-22T18:28:04

Please see below.

We have,

#color(red)((1)tan(A+B)=(tanA+tanB)/(1-tanAtanB)#

Using #(1)# we get ,

#tan70^circ=color(red)(tan(50^circ+20^circ)#

#=>tan70^circ=color(red)((tan50^circ+tan20^circ)/(1- tan50^circtan20^circ#

#=>tan70^circ(1-tan50^circtan20^circ)=tan50^circ+tan20^circ#

#=>tan70^circ-tan70^circtan50^circtan20^circ=tan50^circ+tan20^circ#

#=>tan70^circ-color(blue)(tan70^circ)tan20^circtan50^circ=tan50^circ+tan20^circ#

#tan70^circ-color(blue)(tan(90^circ-20^circ))tan20^circtan50^circ=tan50^circ+tan20^circ#

#=>tan70^circ-color(blue)(cot20^circ)tan20^circtan50^circ=tan50^circ+tan20^circ#

#=>tan70^circ-color(blue)(1)*tan50^circ=tan50^circ+tan20^circ#

#=>tan70^circ=tan50^circ+tan50^circ+tan20^circ#

#=>tan70^circ=2tan50^circ+tan20^circ#

Note:

#color(blue)(tan70^circ=tan(90^circ-20^circ)=cot20^circ#

#andcolor(blue)( tantheta*cottheta=1=>tan20^circcot20^circ=1#