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Question
Find the value of `cos^-1 (1/2) + tan^-1 (1/sqrt(3))`
Solution
Let `cos^-1 (1/2)` = x
∴ cos x = `1/2`
= `cos pi/3`
The principal value branch of cos−1 is [0, π] and `0 ≤ pi/3 ≤ pi`
∴ x = `pi/3`
∴ `cos^-1 1/2 = pi/3`
Let `tan^-1 (1/sqrt(3))` = y
∴ tan y = `1/sqrt(3`
= tan `pi/6`
The principal value branch of tan−1 is `((-pi)/2, pi/2)` and `- pi/2, < pi/6 < pi/2`
∴ y = `pi/6`
∴ `tan^-1 (1/sqrt(3)) = pi/6`
∴ `cos^-1 (1/2) + tan^-1 (1/sqrt(3)) = pi/3 + pi/6`
= `(3pi)/6`
= `pi/2`
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