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If sin [cot−1 (x+1)] = cos(tan−1x), then find x. - Mathematics

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If sin [cot−1 (x+1)] = cos(tan1x), then find x.

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Solution

If sin [cot−1 (x+1)] = cos(tan1x), 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

Concept: Inverse Trigonometric Functions (Simplification and Examples)
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