sin^-1 (x) + cos^-1(x) = pi/2

Let sin^-1 (x) = a ==> sin(a) = x

Let cos^-1 (x) = b ==> cos(b) = x

Then we conclude that:

sin(a) = cos(b)

We need to prove that a+ b= pi/2

We will use the right angle triangle to prove.

Let a , b, and c=90 be the angles of a right angle triangle.

Then we know that

sina = opposite/hypotenuse= bc/ac

cosb= adjacent/ hypotenuse = bc/ac

Then we conclude that sina = cosb

==> But we know that the sum of the angles in a triangle is 180 degrees.

But one of the angles in a right angle triangle is 90 degree.

Then the sum of the other two angles (a and b ) is 180-90 = 90

Then a+ b= 90 = pi/2

==> But sin^-1(x)=a and cos^-1(x) = b

==> sin^-1 (x) + cos^-1 (x) = pi/2..........q.e.d