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The number of real solutions of tan^(-1)...

The number of real solutions of `tan^(-1)sqrt(x(x+1))+sin^(-1)sqrt(x^2+x+1)=pi/2` is

A

0

B

1

C

2

D

`infty`

Text Solution

Verified by Experts

The correct Answer is:
C
Given, `tan^(-1)sqrt(x(x+1))=(pi)/(2)-sin^(-1)sqrt(x^(2)+x+1)`
`cos^(-1)(1)/(sqrt((x^(2)+x)^(2)+1))=cos^(-1)sqrt(x^(2)+x+1)`
`rArr(1)/(sqrt((x^(2)+x)^(2)+1))=sqrt(x^(2)+x+1)`
`rArr1=(x^(2)+x+1)[(x^(2)+x)^(2)+1]`
`rArr(x^(2)+x)^(3)+(x^(2)+x)^(2)+(x^(2)+x)+1=1`
`rArr(x^(2)+x){(x^(2)+x)^(2)+(x^(2)+x)}+1=0`
`rArrx^(2)+x=0rArrx=0,-1`
Thus , number of solutions =2
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