Question

Prove that 2tan 50º = tan 70° - tan20°​

Answer 1

Answer:

tan ( 70 - 20) = tan50

tan70 - tan20 = tan50(1+ tan20 tan70)

tan70 - tan20= tan50 + tan20 tan70 tan50

tan70 - tan20= tan50 + tan( 90 - 70) tan70tan50

tan70 - tan20= tan50 +cot70 *tan70*tan50

tan70 - tan20 = 2tan50

since

tanA *cotA=1

Hence LHS = RHS proved

Answer 2

According to the trigonometric identity,

tan70 = tan (20 + 50)

tan70= (tan20 + tan50) / 1-tan20 tan50

tan70 - tan20 tan50 tan70= tan20 + tan50

Also tan70 tan20 = tan70 cot70 = 1

Hence, it will change to following equation

tan70 - tan50 = tan20 + tan50

So tan70 = tan20 + 2tan50

Complementary angles:

tan70=cot20

tan70tan20=cot20tan20=1

Tangent difference angle formula:

tan(a−b)=tana−tanb1+tanatanb

tan50=tan(70−20)=tan70−tan201+tan70tan20=tan70−tan201+1

2tan50=tan70−tan20

tan70=tan20+2tan50