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Question
Solve the following equation for x:
`tan^-1(2+x)+tan^-1(2-x)=tan^-1 2/3, where x< -sqrt3 or, x>sqrt3`
Solution
We know
`tan^-1x+tan^-1y=tan^-1((x+y)/(1-xy))`
∴ `tan^-1(2+x)+tan^-1(2-x)=tan^-1 2/3`
⇒ `tan^-1((2+x+2-x)/(1-(2+x)xx(2-x)))=tan^-1 2/3`
⇒ `4/(1-4+x^2)=2/3`
⇒ `-6+2x^2=12`
⇒ `2x^2=18`
⇒ `x^2=9`
⇒ `x=+-3`
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