Question

Find the exact value of the composite function show work​

Answer 1

Answer:

$\frac{2x}{\sqrt{4x^{2} -1} }$

Step-by-step explanation:

This function can be written as:

$\frac{1}{cos(sin^{-1}(\frac{1}{2x})) }$

$sin^{-1}(\frac{1}{2x})$ 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

$\sqrt{4x^{2} -1}$

The cosine of the angle is adjacent over hypotenuse, so the cosine is

$\frac{\sqrt{4x^{2} -1}}{2x}$

The secant is the inverse of the cosine, so the answer is

$\frac{2x}{\sqrt{4x^{2} -1} }$