Question

prove that tan70°=tan20°+2tan50°

Answer 1

tan(A+B)=tanA+tanB1−tanA.tanBtan(A+B)=tanA+tanB1−tanA.tanBtanA=cot(90−A)tanA=cot(90−A)tanA.cotA=1tanA.cotA=1

We know,

70=50+20

Taking tangent on both sides

tan70=tan(50+20)tan70=tan(50+20)

Using Prerequisite (1)

tan70=tan50+tan201−tan50.tan20tan70=tan50+tan201−tan50.tan20

tan70.(1−tan50.tan20)=tan50+tan20tan70.(1−tan50.tan20)=tan50+tan20

tan70−tan70.tan50.tan20=tan50+tan20tan70−tan70.tan50.tan20=tan50+tan20

tan70−tan70.tan50.tan(90−70)=tan50+tan20tan70−tan70.tan50.tan(90−70)=tan50+tan20

tan70−tan70.tan(90−70).tan50=tan50+tan20tan70−tan70.tan(90−70).tan50=tan50+tan20

Using Prerequisite (2)

tan70−tan70.cot70.tan50=tan50+tan20tan70−tan70.cot70.tan50=tan50+tan20

Using Prerequisite (3)

tan70−1.tan50=tan50+tan20tan70−1.tan50=tan50+tan20

tan70=2.tan50+tan20

Answer 2



 

Thus, tan 70 = tan 20 + 2 tan 50


Cheers