Question

The value of `tan [1/2 cos^-1 (sqrt(5)/3)]` is ______.

Options

• `(3 + sqrt(5))/2`

• `(3 - sqrt(5))/2`

• `(-3 + sqrt(5))/2`

• `(-3 - sqrt(5))/2`

Answer 1

The value of `tan [1/2 cos^-1 (sqrt(5)/3)]` is `underlinebb((3 - sqrt(5))/2)`.

Explanation:

`x = tan [1/2 cos^-1 (sqrt(5)/3)]`

Let `cos^-1  sqrt(5)/3 = θ`

`cos θ = sqrt(5)/3`

`x = sqrt(1 - sqrt(5)/3)/sqrt(1 + sqrt(5)/3)`

= `sqrt(3 - sqrt(5))/sqrt(3 + sqrt(5))`

= `sqrt(3 - sqrt(5))/sqrt(3 + sqrt(5)) xx sqrt(3 - sqrt(5))/sqrt(3 - sqrt(5))`

= `(3 - sqrt(5))/sqrt((3)^2 - (sqrt(5))^2`

= `(3 - sqrt(5))/sqrt(9 - 5)`

= `(3 - sqrt(5))/2`