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