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Question
Prove the following:
`cos^-1 "x" = tan^-1 (sqrt(1 - "x"^2)/"x")`, if x > 0
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Solution
Let cos-1 x = α
Then, cos α = x, where 0 < α < π
Since, x > 0, 0 < α < `pi/2`
∴ sin α > 0, cos α > 0
Now, `tan^-1 (sqrt(1 - "x"^2)/"x") = tan^-1 (sqrt(1 - cos^2 alpha)/cos alpha)`
`= tan^-1 (sqrt(sin^2 alpha)/cos alpha)`
`= tan^-1 (tan alpha)`
`= alpha = cos^-1 "x"`
Hence, `cos^-1 "x" = tan^-1 (sqrt(1 - "x"^2)/"x")`, if x > 0
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