2arctan(1/2)+arctan(1/7)=?

2 Answers
Jun 24, 2018

2arc tan(1/2)+arc tan (1/7)=arc tan(31/17)

Explanation:

We know that ,

color(red)((1)arc tanx+arc tany=arc tan((x+y)/(1-x*y)) , color(red)(x*y < 1

Let , A=2arc tan(1/2)+arc tan (1/7)

Using (1) we get ,

color(blue)(2arc tan(1/2))=arc tan(1/2)+arc tan(1/2)

color(white)(2arc tan(1/2))=arc tan((1/2+1/2)/(1-1/2*1/2)), x*y=1/2*1/2=1/4<1

color(white)(2arc tan(1/2))=arc tan(1/(1-1/4))

color(white)(2arc tan(1/2))=arc tan(1/(3/4))

color(blue)(2arc tan(1/2)=arc tan(4/3)

So,

A=color(blue)(arc tan (4/3))+arc tan(1/7)tocolor(red)( Apply(1)

A=arc tan((4/3+1/7)/(1-4/3*1/7)) , x*y=4/3*1/7=4/21<1

A=arc tan((28+3)/(21-4))

A=arc tan(31/17)

Jun 24, 2018

61^@26

Explanation:

Use calculator:
tan x = 1/2 --> arc x = 26^@57
2arc x = 2(26.57) = 53^@13
tan y = 1/7 --> arc y = 8^@13
2arctan (1/2) + arctan (1/7) = 53^@13 + 8^@13 = 61^@26