1.

If `tan^(-1)((x-1)/(x-2))+cot^(-1)((x+2)/(x+1))=(pi)/(4)` , find x.

Answer» Given , `"tan"^(-1)(x-1)/(x-2)+"cot"^(-1)(x+2)/(x+1)=(pi)/(4)`
`"tan"^(-1)(x-1)/(x-2)+"tan"^(-1)(x+1)/(x+2)=(pi)/(4)`
`"tan"^(-1)((x-1)/(x-2)+(x+1)/(x+2))/(1(x-1)/(x-2)xx(x+1)/(x+2))=(pi)/(4)`
`((x-1)(x+2)+(x+1)(x-2))/(((x-2)(x+2))/(((x^(2)-4)-(x^(2)-1))/((x-2)(x+2))))="tan"(pi)/(4)`
`(x^(2)+2x-x-2+x^(2)-2x+x-2)/(x^(2)-4-x^(2)+1)=1`
`(2x^(2)-4)/(-3)=1`
`2x^(2)-4=-3`
`2x^(2)=1`
`x^(2)=(1)/(2)`
`x=+-(1)/(sqrt(2))`


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