Prove that sine inverse x = cosec inverse 1/x
Draw the unit circle.
Draw a right triangle in standard position (right angle on the x-axis).
Now label the sides.
The hypotenuse is 1
The height is x
Let angle A be at (0,0)
So, sin(A) = x/1 = x
That is, A = sin^-1(x)
csc(A) = 1/sin(A) = 1/x
So, A = csc^-1(1/x)