Problem 34 - Precalculus with Limits

Question

In Exercises 9-50, verify the identity \( \dfrac{\cos [(\pi/2) - x]}{\sin [(\pi/2) - x]} = \tan x \)

Step-by-Step Solution

Answer

The given identity \( \dfrac{\cos [(\pi/2) - x]}{\sin [(\pi/2) - x]} = \tan x \) is verified by using the co-function identities to simplify the left-hand side of the equation to \( \tan x \), which matches the right-hand side.

Step 1: Trigonometric Co-Function Identities

Use the co-function identities to rewrite the left side expression. These identities state that \( \cos (\pi/2 - x) = \sin x \) and \( \sin (\pi/2 - x) = \cos x \). The left side of the equation can then be rewritten as \( \dfrac{\sin x}{\cos x} \)

Step 2: Simplification

Simplify \( \dfrac{\sin x}{\cos x} \) using the definition of the tangent function. Recall that the tangent of an angle in trigonometry is defined as \( \tan x = \dfrac{\sin x}{\cos x} \). Therefore our expression simplifies to \( \tan x \)

Step 3: Verification of the Identity

The right side of the initial equation was \( \tan x \). After the conversion and simplification steps applied to the left side of the equation, the left side is now also \( \tan x \). Hence, the trigonometric identity is verified.

Key Concepts

Co-Function Identities

Co-function identities in trigonometry are a powerful tool that helps relate different trigonometric functions to one another. These identities are especially useful when dealing with angles that sum up to 90 degrees or \(\pi/2\) radians. Here are the basic co-function relationships:

• \( \cos (\frac{\pi}{2} - x) = \sin x \)
• \( \sin (\frac{\pi}{2} - x) = \cos x \)
• \( \tan (\frac{\pi}{2} - x) = \cot x \)
• \( \cot (\frac{\pi}{2} - x) = \tan x \)

In our exercise, we utilized the first two co-function identities. By substituting, \(\cos (\frac{\pi}{2} - x ) = \sin x\) and \(\sin (\frac{\pi}{2}-x) = \cos x\), we transformed the expression on the left side of the provided identity. This conversion simplifies comparative problems by reducing complex-looking expressions into easily recognizable forms.

Tangent Function

The tangent function is one of the fundamental trigonometric functions, generally denoted as \(\tan x\). It helps in understanding the relationship between the sine and cosine functions. The basic definition of the tangent function is:\[\tan x = \frac{\sin x}{\cos x}\]This definition implies the tangent can be interpreted as the ratio of the opposite side to the adjacent side in a right-angled triangle.
Therefore, in the context of our exercise, simplifying the expression \(\frac{\sin x}{\cos x}\) directly showcases it as \(\tan x\), aligning with the right side of the identity. Understanding the tangent function through this formula is crucial, as it lays the foundation for further identity verification and enhances one's problem-solving skillset in trigonometry.

Verifying Trigonometric Identities

Verifying trigonometric identities involves transforming one side of an equation to match the other using known identities and simplifications. In our exercise, the task is to show that both sides of the given identity \( \frac{\cos [(\pi/2) - x]}{\sin [(\pi/2) - x]} = \tan x \) evaluate to the same expression.The verification process generally follows these steps:

• Use known trigonometric identities, like co-function or reciprocal identities.
• Simplify one or both sides until they appear identical.
• In our case, using co-function identities reduced the left side to \( \tan x \), exactly matching the right side.

Verification not only confirms the legitimacy of an identity but also strengthens understanding of the underlying trigonometric concepts.
It reinforces confidence in applying identities to solve diverse mathematical problems.