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Question
Write the value of tan−1 x + tan−1 `(1/x)` for x < 0.
Solution
`tan^-1x+tan^-1y=tan^-1((x+y)/(1-xy))`
When `x<0,1/x<0,` then both are negative.
Let x = -y, y > 0
Then,
`tan^-1x+tan^-1 1/x=tan^-1 (-y)+tan^-1(-1/y)`
`=-(tan^-1y+tan^-1 1/y)`
`=-tan^-1((y+1/y)/(1-y1/y)), y>0`
`=-tan^-1((y^2+1)/0)`
`=-tan^-1(oo)`
`=-tan^-1(tan pi/2)`
`=pi/2`
`thereforetan^-1x+tan^-1 1/x=-pi/2, x<0`
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