Question

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

Answer 1

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