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
