cot(sec-1(x))

Inside the parentheses is sec-1(x).  That says

"The angle whose secant is x.  Let that angle = q.

So we draw a right triangle including the angle q = sec-1(x).



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 .
3. We label the hypotenuse as the numerator x.
4. We label the adjacent side to q as the denominator 1.

  


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

   c² = a² + b²
   x² = 1² + b²
   x² = 1 + b²
 x²-1 = b²
√x²-1 = b

So we label the opposite side as √x²-1



Now we go back to the original problem:

cot(sec-1(x)), that's the cot(q)

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 
√x²-1, so:

Answer: cot(sec-1(x)) = .

Edwin