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प्रश्न
Solve the following equation for x:
`tan^-1 (x-2)/(x-1)+tan^-1 (x+2)/(x+1)=pi/4`
उत्तर
`tan^-1 (x-2)/(x-1)+tan^-1 (x+2)/(x+1)=pi/4`
⇒ `tan^-1(((x-2)/(x-1)+(x+2)/(x+1))/(1-((x-2)/(x-1))((x+2)/(x+1))))=pi/4` `[tan^-1x+tan^-1y=tan^-1((x+y)/(1-xy))]`
⇒ `((((x-2)(x+1)+(x-1)(x+2))/((x-1)(x+1))))/((((x-1)(x+1)-(x-2)(x+2))/((x-1)(x+1))))=tan (pi/4)`
⇒ `((x-2)(x+1)+(x-1)(x+2))/((x-1)(x+1)-(x-2)(x+2))=1`
⇒ `(x^2-x-2+x^2+x-2)/((x^2-1)-(x^2-4))=1`
⇒ `(2x^2-4)/3=1`
⇒ `2x^2-4=3`
⇒ `2x^2=7`
⇒ `x^2=7/2`
∴ `x=+-sqrt(7/2`
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