Problem 94 - Algebra and Trigonometry

Question

Verify each identity. \(\ln |\sec x|=-\ln |\cos x|\)

Step-by-Step Solution

Answer

The identity \(\ln |\sec x|=-\ln |\cos x|\) is verified

Step 1: Interchange Secant term

Interchange the secant term with its equivalent in terms of cosine: \(\ln |\sec x| = \ln |1/\cos x|\)

Step 2: Apply Logarithmic Identity

Employ the identity \(\ln a^b = b \ln a \) to the expression to make it: \(\ln |1/\cos x| = -\ln |\cos x|\)

Step 3: Final Proof

It can now be seen that the two expressions are equal, thus verifying the identity.

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that relate different trigonometric functions to each other. They are the building blocks for simplifying and solving many mathematical problems involving angles and lengths. Key trigonometric functions include sine, cosine, tangent, secant, cosecant, and cotangent.
Some well-known identities include:

• Pythagorean identities:
  • \(\sin^2 x + \cos^2 x = 1\)
  • \(\tan^2 x + 1 = \sec^2 x\)
• Reciprocal identities:
  • \(\sec x = 1 / \cos x\)
  • \(\csc x = 1 / \sin x\)

Understanding these identities helps in simplifying expressions and proving new equations, like in the provided exercise.They connect different trigonometric functions enabling transformations from one form to another. This flexibility is crucial in higher-level mathematics and scientific applications.

Secant Function

The secant function, denoted as \(\sec x\), is one of the six fundamental trigonometric functions. It is the reciprocal of the cosine function, which is illustrated by the identity \(\sec x = 1/\cos x\).
Unlike cosine, which measures the adjacent side over the hypotenuse in a right triangle, secant is simply the hypotenuse over the adjacent side. This role as a reciprocal often serves well in calculus and analysis to simplify different expressions. Its behavior is analogous yet distinct from cosine, as it shares many periodic properties. However, because it is undefined where cosine is zero, spaces where \(\cos x = 0\) are asymptotes for \(\sec x\) in its graph. This function becomes particularly useful in the context of proving identities that require conversions between functions, as we saw in the original exercise.

Cosine Function

The cosine function, denoted as \(\cos x\), is another foundational trigonometric function. It represents the ratio of the length of the adjacent side of a right triangle to its hypotenuse. Cosine is most familiar in its use for solving problems involving angles and distances. It is periodic with a fundamental cycle of \(2\pi\), repeating its values every complete rotation along the unit circle.
Since it is involved in both the Pythagorean identity and the reciprocal identity for secant, cosine is crucial in transforming and simplifying trigonometric expressions. For example, \(\ln |\sec x|\) in the original exercise is rewritten as \(\ln |1/\cos x|\). Understanding these conversions is vital for simplifying logarithmic expressions or identifying equivalencies in various mathematical domains. Cosine's versatile nature provides endless applications, from solving geometry problems to calculating wave functions in physics.

Absolute Value in Logarithms

Absolute value in logarithms help ensure the argument within the logarithm is non-negative, as logarithms of non-positive numbers are undefined within real numbers. This feature is crucial when transforming and equating expressions involving logs. In the identity \(\ln |1/\cos x| = -\ln |\cos x|\), absolute values maintain validity across the entire domain aside from points where cosine reaches zero.
Using absolute values assures us that inputs which might otherwise lead to undefined behavior, like division by zero, are managed correctly. The absolute value serves a protective role, especially when dealing with reciprocals or negative transformations, as in the negative log identity applied here. By maintaining non-negative arguments through absolute values, mathematicians ensure expressions remain valid and consistent across necessary domains.