Problem 27 - Algebra and Trigonometry

Question

Consider the given equation. (a) Verify algebraically that the equation is an identity. (b) Confirm graphically that the equation is an identity. $$ \frac{\cos x}{\sec x \sin x}=\csc x-\sin x $$

Step-by-Step Solution

Answer

The given equation is an identity, confirmed both algebraically and graphically.

Step 1: Simplify Left Side of the Equation

To start, we simplify the left side of the equation. The given equation is \( \frac{\cos x}{\sec x \sin x} \). The secant and sine can be rewritten in terms of cosine and sine: \( \sec x = \frac{1}{\cos x} \), so \( \sec x \sin x = \frac{\sin x}{\cos x} \). Therefore, the left side becomes \( \frac{\cos x}{\frac{\sin x}{\cos x}} = \frac{\cos x \cdot \cos x}{\sin x} = \frac{\cos^2 x}{\sin x} \).

Step 2: Simplify Right Side of the Equation

Now simplify the right side of the equation, which is \( \csc x - \sin x \). The cosecant can be expressed in terms of sine: \( \csc x = \frac{1}{\sin x} \). Thus, the right side becomes \( \frac{1}{\sin x} - \sin x \). To combine these into a single fraction, we find a common denominator: \( \frac{1 - \sin^2 x}{\sin x} \).

Step 3: Use the Pythagorean Identity

Recognize that \( 1 - \sin^2 x \) is an identity, specifically \( \cos^2 x \). Thus, the expression \( \frac{1 - \sin^2 x}{\sin x} \) simplifies to \( \frac{\cos^2 x}{\sin x} \), matching the simplified left side from Step 1.

Step 4: Conclude the Algebraic Verification

Since both sides of the equation simplify to \( \frac{\cos^2 x}{\sin x} \), the equation \( \frac{\cos x}{\sec x \sin x} = \csc x - \sin x \) is confirmed to be an identity.

Step 5: Confirm Graphically

To confirm this identity graphically, plot the functions \( \frac{\cos x}{\sec x \sin x} \) and \( \csc x - \sin x \) on the same graph using a graphing tool or software. You will observe that the graphs overlap completely for all values in their domain, confirming visually that the identity holds true.

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry. It is derived from the Pythagorean theorem, which relates the sides of a right triangle. The most common Pythagorean identity is:

• \( \sin^2 x + \cos^2 x = 1 \)

This equation shows the intrinsic relationship between the sine and cosine functions. By rearranging it, we can express either sine or cosine in terms of the other. For instance, \( \cos^2 x = 1 - \sin^2 x \). This form is particularly useful in trigonometric simplifications. In the given problem, the identity \( 1 - \sin^2 x = \cos^2 x \) was used to simplify the expression on the right side of the equation.
Using these identities helps simplify complex trigonometric expressions and to verify equations as identities.

Cosecant Function

The cosecant function, represented as \( \csc x \), is the reciprocal of the sine function. It is defined as:

• \( \csc x = \frac{1}{\sin x} \)

This function is less commonly used compared to sine or cosine, but it plays an important role in certain areas of mathematics. The domain of the cosecant function is all real numbers except where \( \sin x = 0 \), because division by zero is undefined.
In our original exercise, the right side of the equation involves \( \csc x - \sin x \). By expressing \( \csc x \) in terms of sine, we could create a common denominator for simplification.
For trigonometric identities and simplification problems, knowing how to convert between functions like sine and cosecant is crucial.

Secant Function

The secant function, symbolized as \( \sec x \), is defined as the reciprocal of the cosine function:

• \( \sec x = \frac{1}{\cos x} \)

Similar to the cosecant function, secant exists for all real numbers except where cosine equals zero, as this would result in division by zero.
In the problem, the given equation had \( \sec x \cdot \sin x \) in the denominator, which was key for simplification. By substituting \( \sec x \) with its reciprocal definition, we can simplify complex expressions.
Understanding the reciprocal relationships of secant and cosecant functions opens up pathways to simplify expressions and solve trigonometric equations effectively.

Trigonometric Simplification

Trigonometric simplification involves rewriting complex trigonometric expressions in a simpler or more familiar form. This helps in verifying identities or solving equations. Key steps include:

• Using basic identities like the Pythagorean identity to substitute terms.
• Converting functions to their reciprocals, such as using \( \csc x = \frac{1}{\sin x} \) or \( \sec x = \frac{1}{\cos x} \).
• Finding common denominators to combine fractions.

In the context of the exercise, both sides of the equation were simplified to the same form. By breaking down the components into basic trigonometric functions and applying the identities, the algebraic pathway was clear.
The ability to simplify trigonometric expressions is a powerful tool in mathematics, making complex problems manageable and understandable.