Question 872079
<pre>
sin<font size = 4>(</font>cot<sup>-1</sup>(x)<font size=4>)</font>

The way to interpret that problem is:

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

Let that angle be <font face="symbol">q</font>.

We know that {{{cotangent}}}{{{""=""}}}{{{(adjacent)/(opposite)}}}

We also know that x can be written as {{{x/1}}}

So we draw a right triangle containing the angle <font face="symbol">q</font>
with the side that is adjacent to <font face="symbol">q</font> having length x
and the side that is opposite to angle <font face="symbol">q</font> be 1.
Like this:

{{{drawing(300,200,-.5,2.5,-.5,1.5,triangle(0,0,2,0,2,1),rectangle(1.8,0,2,.2),
locate(1,0,x),locate(2.04,.55,1), locate(.4,.19,theta) 


)}}}  

We have drawn a right triangle containing an angle whose
cotangent is {{{x/1}}} which is x.

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

hypotenuse<sup>2</sup> = adjacent<sup>2</sup>+opposite<sup>2</sup>

hypotenuse<sup>2</sup> = x<sup>2</sup>+1<sup>2</sup>

hypotenuse = &#8730;<span style="text-decoration: overline">x²+1</span>

{{{drawing(300,200,-.5,2.5,-.5,1.5,triangle(0,0,2,0,2,1),rectangle(1.8,0,2,.2),
locate(1,0,x),locate(2.04,.55,1), locate(.4,.19,theta),
locate(.6,.8,sqrt(x^2+1)) 


)}}} 

Now, we remember that the way to interpret the problem

sin<font size = 4>(</font>cot<sup>-1</sup>(x)<font size=4>)</font>

is:

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

And we let that angle be <font face="symbol">q</font>.

So since {{{sine}}}{{{""=""}}}{{{(opposite)/(hypotenuse)}}} 

then,

sin<font size = 4>(</font>cot<sup>-1</sup>(x)<font size=4>)</font> = sin(<font face="symbol">q</font>) = {{{1/sqrt(x^2+1)}}}.

Edwin</pre>