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Question
The value of the expression `tan (1/2 cos^-1 2/sqrt(5))` is ______.
Options
`2 + sqrt(5)`
`sqrt(5) - 2`
`(sqrt(5) + 2)/2`
`5 + sqrt(2)`
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Solution
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"]`
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