Question

Evaluate `tan (tan^-1(-1) + π/3)`.

Answer 1

We have to find: `tan (tan^-1(-1) + π/3)`

Let’s find tan⁻¹ (−1) first:

tan⁻¹ (−1)

Let y = tan⁻¹ (−1)

We know that

tan⁻¹ (−x) = −tan⁻¹ x

y = −tan⁻¹ (1)

y = `-π/4`

Since the range of tan⁻¹ is `(-π/2,pi/2)`.

Hence, tan⁻¹ (−1) = `-π/4` is correct.

Now, `tan (tan^-1(-1) + π/3)`

= `tan(-π/4 + π/3)`

= `tan(π/3 - π/4)`

Using tan (A − B) = `(tan A - tan B)/(1 + tan A tan B)`

= `(tan (π/3) - tan (π/4))/(1 + tan (π/3) tan (π/4))`

= `(sqrt(3) - 1)/(1 + sqrt(3) xx 1)`

= `(sqrt(3) - 1)/(1 + sqrt(3))`

= `(sqrt(3) - 1)/(sqrt(3) + 1)`