Question

The value of `tan^-1  1/3 + tan^-1  1/5 + tan^-1  1/7 + tan^-1  1/8` is ______.

Options

• `(11 pi)/5`

• `pi/4`

• `pi`

• `(3pi)/4`

Answer 1

The value of `tan^-1  1/3 + tan^-1  1/5 + tan^-1  1/7 + tan^-1  1/8` is `underline(pi/4)`.

Explanation:

We have,

`tan^-1  1/3 + tan^-1  1/5 + tan^-1  1/7 + tan^-1  1/8`

`=> tan^-1 ((1/3 + 1/5)/(1 - 1/3 xx 1/5)) + tan^-1 ((1/7 + 1/8)/(1 - 1/7 xx 1/8))   ...(because tan^-1 x + tan^-1 y = tan^-1 ((x + y)/(1 - "xy")))`

`= tan^-1 (8/14) + tan^-1 (15/55)`

`= tan^-1 (4/7) + tan^-1 (3/11)`

`= tan^-1 ((4/7 + 3/11)/(1 - 4/7 xx 3/11))`

`= tan^-1 ((44 + 21)/(77 - 12))`

`= tan^-1 (65/65)`

`= tan^-1 (1) = pi/4`