Question 807741
<pre>
cot(sec<sup>-1</sup>(x))

Inside the parentheses is sec<sup>-1</sup>(x).  That says

"The angle whose secant is x.  Let that angle = <font face="symbol">q</font>.

So we draw a right triangle including the angle <font face="symbol">q</font> = sec<sup>-1</sup>(x).

{{{drawing(200,200,-.5,2.5,-.5,2.5, triangle(0,0,1.5,0,1.5,2),
locate(.4,.3,theta)

 )}}}

Now to make sure that this angle is the angle whose secant is x,
We do the following:

1. We note that the secant is the hypotenuse over the adjacent. 
2. We notice that x is equivalent to {{{x/1}}}.
3. We label the hypotenuse as the numerator x.
4. We label the adjacent side to <font face="symbol">q</font> as the denominator 1.

  
{{{drawing(200,200,-.5,2.5,-.5,2.5, triangle(0,0,1.5,0,1.5,2),
locate(.3,.3,theta), locate(.75,0,1), locate(.5,1.1,x)

 )}}}

Next we use the Pythagorean theorem to find the opposite side:

   c² = a² + b²
   x² = 1² + b²
   x² = 1 + b²
 x²-1 = b²
&#8730;<span style="text-decoration: overline">x²-1</span> = b

So we label the opposite side as &#8730;<span style="text-decoration: overline">x²-1</span>

{{{drawing(200,200,-.5,2.5,-.5,2.5, triangle(0,0,1.5,0,1.5,2),
locate(.3,.3,theta), locate(.75,0,1), locate(.5,1.1,x),
locate(1.5,1.1,sqrt(x^2-1))

 )}}}

Now we go back to the original problem:

cot(sec<sup>-1</sup>(x)), that's the cot(<font face="symbol">q</font>)

We want the cotangent of that angle.  We know that the cotangent is
the adjacent over the opposite.   The adjacent is 1 and the opposite is 
&#8730;<span style="text-decoration: overline">x²-1</span>, so:

Answer: cot(sec<sup>-1</sup>(x)) = {{{1/sqrt(x^2-1)}}}.

Edwin</pre>