Solve the equation: `tan^(-1)sqrt(x^2+x)+sin^(-1)sqrt(x^2+x+1)=pi/2`

Explanation

`tan ^{-1} sqrt{x^{2}+x}+sin ^{-1} sqrt{x^{2}+x+1}=frac{pi}{2}`

`tan ^{-1} sqrt{x^{2}+x}=frac{pi}{2}-sin ^{-1} sqrt{x^{2}+x+1}`

`tan ^{-1} sqrt{x^{2}+x}=cos ^{-1} sqrt{x^{2}+x+1}`

Apply `tan ^{-1} y=cos ^{-1} frac{1}{sqrt{1+y^{2}}}` on R.H.S.

`cos ^{-1} frac{1}{sqrt{1+(sqrt{x^{2}+x})^{2}}}=cos ^{-1} sqrt{x^{2}+x+1}`

`cos ^{-1} frac{1}{sqrt{x^{2}+x+1}}=cos ^{-1} sqrt{x^{2}+x+1}`

`Rightarrow frac{1}{sqrt{x^{2}+x+1}}=sqrt{x^{2}+x+1}`

`x^{2}+x+1=1`

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