Question

solve: sin-1x+ sin-1(1-x) = cos-1x

Answer 1

the functions inside the Latex script are not being interpreted properly by the Interpreter of the language - today..  there were many other questions - with same difficulty.  So I have changed the answer.  please understand the logic. thanks.

Let
$Sin^{-1}x=A, Sin^{-1}(1-x)=B, Cos^{-1}(x)=C\\x=SinA, 1-x=SinB, x=CosC\\CosA=sqrt{(1-x^2)},\\ CosB=sqrt{(1-(1-x)^2)}=sqrt{(2x-x^2)}\\SinC=sqrt{(1-x^2)}\\\\Given,. A+B=C\\...Sin(A+B)=SinC\\SinACosB+SinBCosA=SinC$

$x* sqrt {(2x-x^2)}+(1-x)* sqrt{(1-x^2)}=sqrt{(1-x^2)}\\ x*sqrt{(2x-x^2)}=x*sqrt{(1-x^2)}\\x=0.. is-a-solution\\2x-x^2=1-x^2\\2x=1\\x=1/2$

So you have two solutions  x =  0 and 1/2.