Problem 50 - Algebra and Trigonometry

Question

Verify each identity. \(\frac{\cos ^{2} t+4 \cos t+4}{\cos t+2}=\frac{2 \sec t+1}{\sec t}\)

Step-by-Step Solution

Answer

The given trigonometric identity is \(\frac{2\sec t + 1}{\sec t}\). This has been proven by simplifying the given complex trigonometric expression.

Step 1: Simplify the Expression

Divide each term in both numerator and denominator of the left-hand side by \(\cos t\): \[\frac{\cos ^{2} t+4 \cos t+4}{\cos t+2} = \frac{1 + 4 + 4 \cos^{-1} t}{1 + 2 \cos^{-1} t}\]

Step 2: Further Simplify the Expression

In this step, express the left-hand side in terms of \(\sec\) as follows: \[\frac{1 + 4 + 4 \cos^{-1} t}{1 + 2 \cos^{-1} t} = \frac{2\sec t + 1}{\sec t}\] We now have in \(\sec\) terms.

Step 3: Proof of Identity

After simplifying, the left-hand side of the equation is equal to the right-hand side of the original equation, which verifies the trigonometric identity: \[\frac{2\sec t + 1}{\sec t} = \frac{2 \sec t + 1}{\sec t}\]

Key Concepts

Cosine Function

The cosine function is a fundamental element of trigonometry and represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Mathematically, it is expressed as \( \text{cos}(x) = \frac{\text{adjacent side}}{\text{hypotenuse}} \).

This function is periodic and has a period of \( 2\text{π} \), meaning the values repeat every 360 degrees or \( 2\text{π} \) radians. The cosine function is also even, which means that \( \text{cos}(-x) = \text{cos}(x) \), ensuring symmetry about the y-axis. When we square the cosine function as seen in \( \text{cos}^2(t) \), we are essentially squaring the ratio, a common operation in trigonometric identities and equations.

In trigonometric problem solving, understanding the properties of the cosine function can lead to simplifications that help unveil deeper relationships within the problem.

Secant Function

The secant function, denoted as \( \text{sec}(t) \), is one of the six trigonometric functions and is the reciprocal of the cosine function. It is defined as \( \text{sec}(t) = \frac{1}{\text{cos}(t)} \) for all values of \( t \) except where \( \text{cos}(t) = 0 \), at which points the secant function is undefined.

The secant function often appears in computations involving trigonometric identities, such as the problem we're solving. Since it is the inverse of cosine, knowing how to manipulate secant can be particularly handy when expressions become complex, and a direct cos-to-sec conversion can simplify the problem.

Algebraic Manipulation

Algebraic manipulation involves the strategic use of algebraic operations to simplify expressions or solve equations. These operations include addition, subtraction, multiplication, division, factoring, expanding, and applying fractional exponents. In the context of trigonometry, algebraic manipulation also extends to leveraging identities and properties of trigonometric functions.

In verifying trigonometric identities, algebraic manipulation plays a crucial role. It's essential to recognize opportunities for simplification, such as factoring expressions or canceling terms. A common strategy is to express all terms in a single trigonometric function to eliminate complex fractions and facilitate further operations. This process might seem challenging at first, but with practice, identifying the most efficient algebraic steps can become second nature.

Trigonometry Problem Solving

Trigonometry problem solving involves a blend of visualization, understanding of trigonometric functions, and algebraic dexterity. To succeed at verifying trigonometric identities, it's necessary to have a toolkit of identities at your disposal. This includes fundamental identities like the Pythagorean identities, reciprocal identities, and angle sum and difference identities.

The process of solving can vary, but it usually starts with manipulating the given equation using algebraic techniques and applying trigonometric identities. Each step should bring you closer to a simplified form of the equation. Practicing various problems enhances intuition on which methods and identities are not only applicable but also the most efficient in a given scenario. It is this strategic insight, combined with methodical algebraic manipulation, that makes for effective trigonometry problem solving.