Problem 51 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\frac{\cot x \tan x}{\cos x}=\sec x$$

Step-by-Step Solution

Answer

The given expression \(\frac{\cot x \tan x}{\cos x}\) simplifies to \(1/\cos x\) which is equal to \(\sec x\), thereby verifying the identity.

Step 1: Express in terms of base identities

Rewrite cotangent and tangent in terms of sine and cosine and simplify. Cotangent \(x\) can be written as \(\frac{1}{\tan x}\), or \(\frac{\cos x}{\sin x}\) and tangent \(x\) as \(\frac{\sin x}{\cos x}\). Substitute these into the equation:\n\(\frac{(\cos x/\sin x)(\sin x/\cos x)}{\cos x}\)

Step 2: Simplification

After substitution, simplify the equation by cancelling out common terms:\n\(=\frac{\cos x}{\cos x}=\) 1. The expression simplifies to 1 divided by cosine \(x\).

Step 3: Show equivalence

We know that secant \(x = 1/\cos x\). The simplified equation is also \(1/\cos x\) which is equal to secant \(x\). This confirms that both sides of the equation are equal, verifying the identity.

Step 4 (Optional): Verification with a graphing utility

Plot \(y = \frac{\cot x * \tan x}{\cos x}\) and \(y = \sec x\) in the graphing utility. If the graphical forms overlap completely, this indicates that both expressions are identical in all values of x where they are defined, serving as a visual verification of the identity.

Key Concepts

Cotangent

In trigonometry, cotangent is a key function that is symbolized as \( \cot x \). It relates to the other main trigonometric functions, sine and cosine. Cotangent of an angle is defined as the ratio of the adjacent side to the opposite side in a right triangle: \[ \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \] This transformation is useful when verifying trigonometric identities, as it can simplify equations when expressed in terms of sine and cosine. For instance, in our identity, expressing \( \cot x \) as \( \frac{\cos x}{\sin x} \) allows us to pair it neatly with \( \tan x \), simplifying further calculations. Understanding how cotangent behaves with other functions is essential. For example, multiplying \( \cot x \) by \( \tan x \) results in 1, which is a powerful tool in simplifying complex identities. Also, it should be noted that cotangent is undefined when \( \tan x \) is zero, or when \( \sin x \) is zero, typically at multiples of \( \pi \).

Tangent

Tangent, denoted by \( \tan x \), is another fundamental trigonometric function. It is defined as the ratio of the opposite side to the adjacent side in a right triangle: \[ \tan x = \frac{\sin x}{\cos x} \] This expression makes tangent an easy function to work with when we are dealing with equations involving sine and cosine. In the identity \( \frac{\cot x \tan x}{\cos x} = \sec x \), \( \tan x \) becomes essential as it balances out the cotangent, resulting in simplification to \( \cos x \). Tangent function exhibits interesting properties, such as periodicity and undefined points. The periodicity of \( \tan x \) means it repeats every \( \pi \) radians, and it is undefined wherever \( \cos x = 0 \). Using tangent with other trigonometric functions like cotangent and secant can help unravel or verify trigonometric identities, offering a robust framework for solving complex trigonometric equations.

Secant

Secant is represented as \( \sec x \) and is the reciprocal of the cosine function. It is mathematically expressed as: \[ \sec x = \frac{1}{\cos x} \] Understanding this form is crucial, especially in simplifying and verifying trigonometric identities. In the exercise you provided, the identity \( \frac{\cot x \tan x}{\cos x} = \sec x \) simplifies to \( \sec x \), thus proving that the two expressions are indeed equivalent. Secant reveals valuable characteristics in trigonometry. It is undefined wherever \( \cos x = 0 \) because dividing by zero is undefined. These points occur at odd multiples of \( \frac{\pi}{2} \). Using secant in identities helps bridge connections between different trig functions and makes complex trigonometric problems more approachable.