Question

`tan^(-1) sqrt3 - sec^(-1)(-2)` is equal to ______.

Options

• π

• `-pi/3`

• `pi/3`

• `(2pi)/3`

Answer 1

 `tan^(-1) sqrt3 - sec^(-1)(-2)` is equal to `underlinebb(-pi/3)`.

Explanation:

We know that the range of principal value branch of tan⁻¹ and sec⁻¹ are `(-pi/2, pi/2)` and [0, π] respectively.

Let `tan^(-1) sqrt3` = x ⇒ `sqrt3` = tan x

Then `sqrt3 = tan(pi/3)`

Where `pi/3 ∈ (-pi/2, pi/2)`

Let sec⁻¹ (−2) = y ⇒ −2 = sec y

Then, −2 = `sec(pi/3)`

= `sec(pi - pi/3) = sec  (2pi)/3`

Where `(2pi)/3 ∈ [0, pi] - {pi/2}`

∴ `tan^(-1) sqrt3 - sec^(-1)(-2)`

= `pi/3 - (pi - pi/3)`

= `pi/3 - (2pi)/3`

= `-pi/3`