Question

How do you simplify `tan[cos^-1(-(sqrt3/2))]`?

Answer 1

`- 1/sqrt 3`.

Explanation:

The cosine value in brackets being negative,

`cos^ (-1)(-sqrt3/2) in [pi/2, pi ]`. So,

`tan cos^ (-1)(-sqrt3/2)`

`= tan (cos^ (-1)cos (pi-pi/6))`

`= tan (pi - pi/6)`

`= -tan (pi/6)`

`=-1/sqrt 3`

Answer 2

-0.5774

Explanation:

There is an Inverse Trigonometric Identity that will help on this webpage
https://brilliant.org/wiki/inverse-trigonometric-identities/

It says that `cos^(-1)(-x) = pi - cos^(-1)(x)`

Using that, your expression can be simplified to

`tan(pi - cos^-1(sqrt(3)/2))`

Do you need to take it further? Taking just part of the above

`cos^-1(sqrt(3)/2) = pi/6`

Plugging that into `tan(pi - cos^-1(sqrt(3)/2))` we get

`tan(pi - pi/6) = tan((5pi)/6)`

Using my calculator, the whole thing is equal to -0.5774...

I hope this helps,
Steve