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If x lt 0, the prove that cos^(-1) ((1 +...

If `x lt 0`, the prove that `cos^(-1) ((1 + x)/(sqrt(2(1 + x^(2))))) = (pi)/(4) - tan^(-1) x`

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`f(x) = cos^(-1) ((1 + x)/(sqrt(2 (1 + x^(2))))), x le 0`
Putting `x = tan theta`, where `theta in (-(pi)/(2),0]`, we get
`f(x) = cos^(-1) ((1 + tan theta)/(sqrt(2 sec^(2) theta)))`
`= cos^(-1) ((1)/(sqrt2) (cos theta + sin theta))`
`= cos^(-1) (cos (theta - (pi)/(4)))`
Now, `theta - (pi)/(4) in ((-3pi)/(4), (-pi)/(4)]`
`rArr f(x) = - (theta - (pi)/(4)) = (pi)/(4) - tan^(-1) x`
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