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Prove that cos (tan^(-1) (sin (cot^(-1) ...

Prove that `cos (tan^(-1) (sin (cot^(-1) x))) = sqrt((x^(2) + 1)/(x^(2) + 2))`

Text Solution

Verified by Experts

`E = cos (tan^(-1) (sin (cot^(-1) x)))`
Let `cot^(-1) x = theta`

`:. E = cos [tan^(-1) (sin (sin^(-1).(1)/(sqrt(1 + x^(2))))] " if " x gt 0`
or `E = cos [tan^(-1) (sin (pi - sin^(-1).(1)/(sqrt(1 + x^(2)))))] " if " x lt 0`
In each case, `E = cos [tan^(-1) (1)/(sqrt(1 + x^(2)))]`
`:. E = cos [cos^(-1) sqrt((1 + x^(2))/(2 + x^(2)))]`
`= sqrt((x^(2) + 1)/(x^(2) + 2))`
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