sin(cot-1(x))

The way to interpret that problem is:

"Find the sine of the angle whose cotangent is x."

Let that angle be q.

We know that 

We also know that x can be written as 

So we draw a right triangle containing the angle q
with the side that is adjacent to q having length x
and the side that is opposite to angle q be 1.
Like this:

  

We have drawn a right triangle containing an angle whose
cotangent is  which is x.

Now we calculate the hypotenuse by the Pythagorean theorem,
because we'll need that to find the sine.

hypotenuse2 = adjacent2+opposite2

hypotenuse2 = x2+12

hypotenuse = √x²+1

 

Now, we remember that the way to interpret the problem

sin(cot-1(x))

is:

"Find the sine of the angle whose cotangent is x."

And we let that angle be q.

So since  

then,

sin(cot-1(x)) = sin(q) = .

Edwin