Analyzing the Trigonometric Function f(α)

The question asks us to simplify an expression involving a function \(f(\alpha)\) defined as \(f(\alpha) = \sqrt{\sec^2\alpha-1}\) and then compare the simplified expression to given options which are also in the form of \(f\) applied to combinations of \(\alpha\) and \(\beta\).

First, let's simplify the function \(f(\alpha)\). We know a fundamental trigonometric identity:

\(\sec^2\alpha - \tan^2\alpha = 1\)

Rearranging this identity, we get:

\(\sec^2\alpha - 1 = \tan^2\alpha\)

Now substitute this into the definition of \(f(\alpha)\):

\(f(\alpha) = \sqrt{\tan^2\alpha}\)

The square root of a square is the absolute value. So, \(f(\alpha) = |\tan\alpha|\). However, the structure of the expression we need to evaluate, \(\frac{f(\alpha)+f(\beta)}{1-f(\alpha) f(\beta)}\), strongly suggests the tangent addition formula \(\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\). For this formula to match directly, it implies that in the context of this problem, we should consider \(f(\alpha) = \tan\alpha\) and \(f(\beta) = \tan\beta\), potentially assuming angles are in quadrants where tangent is positive or that \(\sqrt{\tan^2 x} = \tan x\) is used for simplicity within this specific problem structure.

Assuming \(f(\alpha) = \tan\alpha\) and \(f(\beta) = \tan\beta\), let's evaluate the given expression:

Expression = \(\frac{f(\alpha)+f(\beta)}{1-f(\alpha) f(\beta)}\)

Substitute \(f(\alpha) = \tan\alpha\) and \(f(\beta) = \tan\beta\):

Expression = \(\frac{\tan\alpha + \tan\beta}{1 - \tan\alpha \tan\beta}\)

Applying the Tangent Addition Formula

The expression \(\frac{\tan\alpha + \tan\beta}{1 - \tan\alpha \tan\beta}\) is a standard trigonometric identity. It is the formula for the tangent of the sum of two angles:

\(\tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha \tan\beta}\)

Therefore, the given expression is equal to \(\tan(\alpha + \beta)\).

Comparing with the Options

Now let's look at the options and see which one matches our result. The options are given in terms of the function \(f\).

Using our simplified form \(f(x) = \tan x\), let's write out what each option represents:

1. \(f(\alpha - \beta)\): This would be \(\tan(\alpha - \beta)\)
2. \(f(\alpha + \beta)\): This would be \(\tan(\alpha + \beta)\)
3. \(f(\alpha) f(\beta)\): This would be \((\tan\alpha)(\tan\beta)\)
4. \(f(\alpha\beta)\): This would be \(\tan(\alpha\beta)\)

Our simplified expression is \(\tan(\alpha + \beta)\), which exactly matches the form of option 2, \(f(\alpha + \beta)\).

Thus, \(\frac{f(\alpha)+f(\beta)}{1-f(\alpha) f(\beta)}\) is equal to \(f(\alpha + \beta)\).

Conclusion

By simplifying the function \(f(\alpha)\) and evaluating the given expression, we found that it equals \(\tan(\alpha + \beta)\), which is equivalent to \(f(\alpha + \beta)\) under the likely intended interpretation of \(f(x) = \tan x\).

Revision Table: Key Trigonometric Identities

Additional Information: Understanding \(\sqrt{\tan^2\alpha}\)

It's important to note that \(\sqrt{x^2} = |x|\). So, \(\sqrt{\tan^2\alpha} = |\tan\alpha|\). If we were to use the absolute value, the expression would be \(\frac{|\tan\alpha|+|\tan\beta|}{1-|\tan\alpha| |\tan\beta|}\). This form does not directly simplify to \(|\tan(\alpha+\beta)|\) because the absolute value function does not distribute over addition and subtraction in the same way. For example, \(|a+b| \neq |a|+|b|\) in general. However, given the options are standard trigonometric sum formulas, the problem implicitly guides us to use the interpretation where \(f(\alpha) = \tan\alpha\), which is valid if \(\tan\alpha > 0\), or if we consider the identity in a context where the sign naturally aligns or is handled by the structure of the sum formula itself.

In many problems involving trigonometric identities and formulas like \(\tan(A+B)\), when simplifying \(\sqrt{\tan^2\alpha}\) derived from \(\sec^2\alpha-1\), the context (like the presence of addition formula structures in options) often implies working with \(\tan\alpha\) directly, assuming the principal values or regions where tangent is positive, or simply using \(\tan\alpha\) as the functional form for the purpose of applying identities.