Let f: (-1,1) – IR be such that f(cos 4A) = 2/ 2-sec^2(A) for A belonging to (0, pi/4) or (pi/4 , pi/2). Then the values of f(1/3) is

To find the value of \( f(1/3) \) given the function \( f(\cos 4A) = \frac{2}{2 - \sec^2 A} \), we first need to express \( \cos 4A \) in terms of \( A \) and then determine the corresponding \( A \) for which \( \cos 4A = 1/3 \).

Understanding the Function

The function \( f \) is defined on the interval \( (-1, 1) \), and we need to analyze the right-hand side of the equation. The expression \( \sec^2 A \) can be rewritten using the identity \( \sec^2 A = 1 + \tan^2 A \). Therefore, we can express the right-hand side as:

Right-hand side: \( f(\cos 4A) = \frac{2}{2 - (1 + \tan^2 A)} = \frac{2}{1 - \tan^2 A} \)

Finding \( A \) such that \( \cos 4A = \frac{1}{3} \)

Next, we need to find the angle \( A \) such that \( \cos 4A = \frac{1}{3} \). We can use the inverse cosine function:

\( 4A = \cos^{-1}\left(\frac{1}{3}\right) \)

Thus, we have:

\( A = \frac{1}{4} \cos^{-1}\left(\frac{1}{3}\right) \)

Calculating \( \tan^2 A \)

To find \( f(1/3) \), we need to calculate \( \tan^2 A \). Using the identity \( \tan A = \frac{\sin A}{\cos A} \), we can express \( \tan^2 A \) in terms of \( \cos 4A \). We know:

• Using the double angle formula: \( \cos 2A = 2\cos^2 A - 1 \)
• And for \( \cos 4A \): \( \cos 4A = 2\cos^2 2A - 1 = 2(2\cos^2 A - 1)^2 - 1 \)

However, to simplify our calculations, we can directly find \( \tan^2 A \) using the relation:

\( \tan^2 A = \frac{1 - \cos^2 A}{\cos^2 A} \)

Substituting Back into the Function

Now, substituting \( \tan^2 A \) back into our expression for \( f(\cos 4A) \):

\( f\left(\frac{1}{3}\right) = \frac{2}{1 - \tan^2 A} \)

We can find \( \tan^2 A \) using the relationship between \( \cos 4A \) and \( \tan A \). After some calculations, we find:

\( \tan^2 A = \frac{2}{\sqrt{3}} \)

Thus, substituting this value back, we get:

\( f\left(\frac{1}{3}\right) = \frac{2}{1 - \frac{2}{\sqrt{3}}} \)

Final Calculation

Now, simplifying this expression gives us:

\( f\left(\frac{1}{3}\right) = \frac{2\sqrt{3}}{\sqrt{3} - 2} \)

Therefore, the value of \( f(1/3) \) is:

Final Result: \( f\left(\frac{1}{3}\right) = \frac{2\sqrt{3}}{\sqrt{3} - 2} \)