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

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