Question

The value of the expression `tan (1/2 cos^-1  2/sqrt(5))` is ______.

पर्याय

• `2 + sqrt(5)`

• `sqrt(5) - 2`

• `(sqrt(5) + 2)/2`

• `5 + sqrt(2)`

Answer 1

The value of the expression `tan (1/2 cos^-1  2/sqrt(5))` is `sqrt(5) - 2`.

Explanation:

We have, `tan (1/2 cos^-1  2/sqrt(5))` 

Let θ = `1/2 cos^-1  2/sqrt(5)`

⇒ 2θ = `cos^-1  2/sqrt(5)`

⇒ cos 2θ = `2/sqrt(5)`

⇒ `(1 - tan^2 theta)/(1 + tan^2 theta) = 2/sqrt(5)`  ......`[because cos 2theta = (1 - tan^2 theta)/(1 + tan^2 theta)]`

⇒ `2 + 2 tan^2 theta = sqrt(5) - sqrt(5) tan^2 theta`

⇒ `sqrt(5) tan^2 theta + 2 tan^2 theta = sqrt(5) - 2`

⇒ `(sqrt(5) + 2) tan^2 theta = sqrt(5) - 2`

⇒ tan²θ = `((sqrt(5) - 2)(sqrt(5) - 2))/((sqrt(5) + 2)(sqrt(5) - 2))` 

⇒ tan²θ = `(sqrt(5) - 2)^2/(5 - 4)`

⇒ tan²θ = `+- (sqrt(5) - 2)`

⇒ tan²θ = `sqrt(5) - 2, [-(sqrt(5) - 2) "is not required"]`