Question

`sin (tan^-1  4/5 + tan^-1  4/3 + tan^-1  1/9 - tan^-1  1/7)` is equal to ______.

Options

• `1/2`

• `1/sqrt2`

• `sqrt3/2`

• 1

Answer 1

`sin (tan^-1  4/5 + tan^-1  4/3 + tan^-1  1/9 - tan^-1  1/7)` is equal to 1.

Explanation:

`sin (tan^-1  4/5 + tan^-1  4/3 + tan^-1  1/9 - tan^-1  1/7)`

= `sin[(tan^-1  4/5 + tan^-1  1/9) + (tan^-1  4/3 - tan^-1  1/7)]`

= `sin[tan^-1  ((4/5 + 1/9)/(1 - 4/5 1/9)) + tan^-1  ((4/3 - 1/7)/(1 + 4/3 . 1/7))]`

= `sin[tan^-1 ((36 + 5)/(45 - 4)) + tan^-1((28 - 3)/(21 + 4))]`

= `sin(tan^-1  41/41 + tan^-1  25/25)`

= `sin[tan^-1 (1) + tan^-1 (1)]`

= `sin(pi/4 + pi/4)`

= `sin  pi/2`

= 1