Question

If \(\mathrm{f}(\mathrm{x})=\cos (\log \mathrm{x})\) then \(\mathrm{f}(\mathrm{x}) \cdot \mathrm{f}(\mathrm{y})-(1 / 2)[\mathrm{f}(\mathrm{x} / \mathrm{y})+\mathrm{f}(\mathrm{xy})]\) has the value (a) 1 (b) \(-1\) (c) 0 (d) 3

Short answer

(c) 0

Step-by-step solution

Express the terms using the given function

First, we need to express the terms of the expression using the function \(f(x) = \cos(\log x)\): $$f(x) = \cos(\log x)$$ $$f(y) = \cos(\log y)$$ $$f(x/y) = \cos(\log(x/y))$$ $$f(xy) = \cos(\log(xy))$$

Plug the terms into the expression

Next, plug these terms into the given expression: $$f(x) \cdot f(y) - \frac{1}{2}[f(x/y) + f(xy)]$$ $$ = (\cos(\log x))(\cos(\log y)) - \frac{1}{2}[\cos(\log(x/y)) + \cos(\log(xy))]$$

Simplify the terms using logarithm properties

Then, simplify the terms using the properties of logarithms. In particular, we will use the properties \(\log (a/b) = \log a - \log b\) and \(\log(ab) = \log a + \log b\): $$ = (\cos(\log x))(\cos(\log y)) - \frac{1}{2}[\cos(\log x - \log y) + \cos(\log x + \log y)]$$

Apply the sum-to-product and product-to-sum trigonometric identities

Now, we will apply the sum-to-product and product-to-sum trigonometric identities to simplify the expression further: $$ = (\cos(\log x))(\cos(\log y)) - \frac{1}{2}\Big[\frac{\cos(\log x) \cos(\log y) - \sin(\log x) \sin(\log y) }{2} + \frac{\cos(\log x) \cos(\log y) + \sin(\log x) \sin(\log y) }{2}\Big]$$

Combine and simplify the terms

Finally, we can combine the terms and simplify the expression: $$ = (\cos(\log x))(\cos(\log y)) - \frac{1}{2}\Big[\frac{2\cos(\log x)\cos(\log y)}{2}\Big]$$ $$ = (\cos(\log x))(\cos(\log y)) - \cos(\log x)\cos(\log y) = 0$$ Thus, the value of the given expression is \(0\). So the answer is: (c) 0

Key concepts

Logarithm Properties

Understanding logarithms is crucial when dealing with exponential functions. Logarithms, often written as \( \log \) followed by the base and the argument, can be simplified using a set of properties. One fundamental property is that the logarithm of a quotient is the difference of the logarithms: \( \log(a/b) = \log a - \log b \). Conversely, the logarithm of a product is the sum of the logarithms: \( \log(ab) = \log a + \log b \). These properties allow us to simplify complex logarithmic expressions, making it easier to incorporate them into trigonometric identities or other mathematical operations. Moreover, knowing that these operations preserve the original relationships between numbers is fundamental when transforming functions and expressions.

If we consider the exercise mentioned, using these logarithm properties allows us to simplify the function's argument from a logarithm of a product or quotient to more manageable separate logarithms.

Sum-to-Product Identity

The sum-to-product identities are essential tools in trigonometry, helping to simplifying trigonometric sums into products. They take the form \(\cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right)\cos\left(\frac{A - B}{2}\right)\) and \(\sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right)\cos\left(\frac{A - B}{2}\right)\). These identities are particularly useful when simplifying expressions involving the sum of cosine or sine functions, which can often arise in the context of waves, oscillations, or when dealing with phasors in electrical engineering.

In the exercise, applying these identities could convert a sum of two cosines into a product involving the sum and difference of their arguments, leading to further simplification.

Product-to-Sum Identity

Conversely, the product-to-sum identities allow for the reverse process, transforming the product of sines and cosines into a sum. These identities have forms such as \(\cos A \cos B = \frac{1}{2}[\cos(A - B) + \cos(A + B)]\) and \(\sin A \sin B = \frac{-1}{2}[\cos(A + B) - \cos(A - B)]\). These transformations are invaluable in various fields of mathematics and physics, including signal processing and communication systems.

In our exercise, we were guided through the elegant transition from the product of two cosine terms into the sum, for which these identities were indispensable. By applying the product-to-sum identity to \(f(x) \cdot f(y)\), we were able to write it in terms of sums, which facilitated the combination and elimination of terms.

Functions and Expressions Simplification

Simplification of functions and expressions is a vital skill in mathematics that involves reducing complexity while retaining equivalency. This process can involve factoring, combining like terms, and employing various algebraic identities. Simplification is not just about making an expression shorter; it is about making it more comprehensible and easier to manipulate.

In the provided exercise, simplification was the final step after application of logarithm properties and trigonometric identities. It involved combining like terms and cancelling out terms when possible. The end goal was to arrive at the simplest form of the original expression, which allowed us to identify that the function's value was zero. Employing these simplification techniques is essential for students to develop a clear and intuitive understanding of complex mathematical concepts.