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Question
If sin [cot−1 (x+1)] = cos(tan−1x), then find x.
Solution
If sin [cot−1 (x+1)] = cos(tan−1x), then find x.
`=>sin{sin^(-1) (1/(sqrt(1+(1+x)^2)))}`
`=cos{cos^(-1)(1/sqrt(1+x^2))} [because cot^(-1)=sin^(-1)1/sqrt(1+x^2) and tan^(-1)x=cos^(-1)(1/sqrt(1+x^2))]`
`⇒1/sqrt(1+(x+1)^2)=1/sqrt(1+x^2)`
`⇒1/sqrt(2+x^2+2x)=1/sqrt(1+x^2)`
`⇒sqrt(1+x2)=sqrt(x^2+2x+2)`
Squaring both sides, we get
⇒1+x2=x2+2x+2
⇒2x+2=1
⇒x=−1/2
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