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प्रश्न
Solve the following equation for x:
tan−1(x −1) + tan−1x tan−1(x + 1) = tan−13x
उत्तर
We know
`tan^-1x+tan^-1y=tan^-1((x+y)/(1-zy))and tan^-1x-tan^-1y=tan^-1((x-y)/(1+xy))`
∴ tan−1(x −1) + tan−1x tan−1(x + 1) = tan−13x
⇒ `tan^-1{(x+1+x-1)/(1-(x+1)xx(x+1))}=tan^-1 3x-tan^-1x`
⇒ `tan^-1((2x)/(2-x^2))(=tan^-1((3x-x)/(1+3x^2))`
⇒ `(2x)/(2-x^2)=(2x)/(1+3x^2)`
⇒ `2-x^2=1+3x^2`
⇒ 4x2 - 1 = 0
⇒ `x^2=1/4`
⇒ `x=+-1/2`
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