Question

`tan(2tan^-1  1/5 + sec^-1  sqrt(5)/2 + 2tan^-1  1/8)` is equal to ______.

Options

• 1

• 2

• `1/4`

• `5/4`

Answer 1

`tan(2tan^-1  1/5 + sec^-1  sqrt(5)/2 + 2tan^-1  1/8)` is equal to 2.

Explanation:

Given expression

`tan(2tan^-1  1/5 + sec^-1  sqrt(5)/2 + 2tan^-1  1/8)` 

`tan(2(tan^-1(1/5) + tan^-1(1/8)) + sec^-1  sqrt(5)/2)`

= `tan(2tan^-1((1/8 + 1/5)/(1 - 1/40)) + sec^-1  sqrt(5)/2)`

Let `sec^-1  sqrt(5)/2` = θ,

Then, tan θ = `1/2`

θ = `tan^-1  1/2`

So, `tan(2tan^-1  1/3 + tan^-1  1/2)`

= `tan(tan^-1((2/3)/(8/9)) + tan^-1  1/2)`

= `tan(tan^-1(3/4) + tan^-1(1/2))`

= `tan(tan^-1((3/4 + 1/2)/(1 - 3/4 xx 1/2)))`

= tan(tan^–1 (2))

= 2