Find the real solution of: tan-1 root(x(x+1)) + sin-1root(x2+x+1) = pi/2.

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Please find below the solution to the asked query:

We have:tan-1xx+1+sin-1x2+x+1=π2tan-1x2+x+sin-1x2+x+1=π2 ;iLet tan-1x2+x=θx2+x=tanθsecθ=1+tan2θ=1+x2+xsecθ=1+x2+xcosθ=11+x2+xθ=cos-111+x2+xHence i becomes:cos-111+x2+x+sin-1x2+x+1=π2 ;iiBut we know thatcos-1α+sin-1α=π2Hence ii is possible  when:11+x2+x=x2+x+1x2+x+12=1x2+x+1=1x2+x=1xx+1=0x=0 or x=-1 Answer

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