Answer:
[tex]\frac{2x}{\sqrt{4x^{2} -1} }[/tex]
Step-by-step explanation:
This function can be written as:
[tex]\frac{1}{cos(sin^{-1}(\frac{1}{2x})) }[/tex]
[tex]sin^{-1}(\frac{1}{2x})[/tex] means the angle whose sine is 1/(2x). This implies that the denominator is the cosine of the angle whose sine is 1/(2x)
Think of a right triangle and consider the angle with this sine. Then the opposite side has length 1 and the hypotenuse has length 2x. The Pythagorean theorem then implies that the adjacent side is
[tex]\sqrt{4x^{2} -1}[/tex]
The cosine of the angle is adjacent over hypotenuse, so the cosine is
[tex]\frac{\sqrt{4x^{2} -1}}{2x}[/tex]
The secant is the inverse of the cosine, so the answer is
[tex]\frac{2x}{\sqrt{4x^{2} -1} }[/tex]
