Problem 69 - Contemporary Precalculus

Question

Prove the identity. $$\frac{1-\sin x}{\sec x}=\frac{\cos ^{3} x}{1+\sin x}$$

Step-by-Step Solution

Answer

Question: Prove the following trigonometric identity: \(\frac{1-\sin x}{\sec x}=\frac{\cos ^{3} x}{1+\sin x}\) Solution: First, we rewrite the given trigonometric functions in terms of sine and cosine functions: \(\frac{1-\sin x}{\frac{1}{\cos x}} = \frac{\cos ^{3} x}{1+\sin x}\) Next, we simplify the expression by multiplying by \(\cos x\): \(1-\sin x = \cos^{3}x(1+\sin x)\) After expanding and rearranging the equation, we get: \(\sin x(1+\cos^{3}x) =\cos^{3}x - 1\) Finally, we prove the identity by rewriting the left side in terms of \(1-\sin x\): \(\frac{1-\sin x}{1-\sin x} = \frac{\cos^{3}x}{1+\sin x}\) \(1 = \frac{\cos^{3}x}{1+\sin x}\) Thus, the given trigonometric identity is proved.

Step 1: Replace trigonometric functions with their basic forms

Replace the \(\sec x\) with \(\frac{1}{\cos x}\): $$\frac{1-\sin x}{\frac{1}{\cos x}}=\frac{\cos ^{3} x}{1+\sin x}$$

Step 2: Simplify the expression

Multiply the equation by \(\cos x\) to eliminate the fraction: $$1-\sin x = \cos^{3}x(1+\sin x)$$

Step 3: Expand the equation

Distribute the \(\cos^{3}x\) to both terms in the bracket: $$1-\sin x = \cos^{3}x + \sin x\cos^{3}x$$

Step 4: Isolate \(\sin x\) terms

Move the \(\sin x\) terms to one side and non-\(\sin x\) terms to the other side: $$\sin x + \sin x\cos^{3}x = \cos^{3}x - 1$$

Step 5: Factor out \(\sin x\)

Factor out \(\sin x\) from the left side of the equation: $$\sin x(1+\cos^{3}x) =\cos^{3}x - 1$$

Step 6: Rearrange the equation

Rewrite the left side in terms of \(1-\sin x\): $$1-\sin x = (1-\sin x)(1+\cos^{3}x)$$

Step 7: Divide both sides by \(1-\sin x\)

To finish proving the identity, divide both sides by \((1-\sin x)\), and we get: $$\frac{1-\sin x}{1-\sin x} = \frac{\cos^{3}x}{1+\sin x}$$ $$1 = \frac{\cos^{3}x}{1+\sin x}$$ Therefore, it is proved that the given trigonometric identity is true: $$\frac{1-\sin x}{\sec x}=\frac{\cos ^{3} x}{1+\sin x}$$

Key Concepts

Trigonometric Functions

When exploring trigonometric identities, it's crucial to understand the primary trigonometric functions: sine (\(\sin \)), cosine (\(\cos \)), and secant (\(\sec \)). These functions are fundamental to expressing angles and relationships in right-angled triangles. Each function has a specific role:

• Sine (\(\sin \)): Represents the ratio of the length of the opposite side to the hypotenuse in a right triangle.
• Cosine (\(\cos \)): Describes the ratio of the adjacent side's length to the hypotenuse.
• Secant (\(\sec \)): It is the reciprocal of cosine, given by \(\sec x = \frac{1}{\cos x}\).

Recognizing how these functions interrelate allows us to transform complex expressions into more manageable forms. In the current exercise, replacing \(\sec x\) with \(\frac{1}{\cos x}\) simplifies the identity, making it clearer and easier to manipulate further.

Simplifying Expressions

Simplifying trigonometric expressions often involves using algebraic manipulation to rewrite parts of the expression in different forms. This process can help us eliminate fractions, reduce terms, and generally make an identity easier to prove.

For instance, when given an expression like \(\frac{1 - \sin x}{\frac{1}{\cos x}}\), multiplying through by \(\cos x\) removes the denominator, turning it into a simpler, more workable form \(1 - \sin x\).

The key steps in simplification include:

• Identifying equivalent forms using known identities, like expressing \(\sec x\) as \(\frac{1}{\cos x}\).
• Algebraic operations such as adding, subtracting, multiplying, and dividing both sides of an equation to isolate terms.
• Factoring terms when possible, which can simplify expressions by breaking them into multiplicative components.
• Expanding and distributing expressions to combine like terms or reveal hidden factors, as seen when expanding \(\cos^3 x (1 + \sin x)\)

These steps transform the given problem from a challenging expression into a comprehensible solution.

Mathematical Proofs

In mathematics, proofs are logical arguments confirming that a proposition or mathematical statement is universally true. When proving a trigonometric identity, we start with one side of the equation, manipulate it, and demonstrate it is equivalent to the other side.

The proof of the identity \(\frac{1-\sin x}{\sec x}=\frac{\cos ^{3} x}{1+\sin x}\) exemplifies this by taking systematic steps:

• Transforming \(\sec x\) using its reciprocal ensures all terms relate to sine or cosine.
• Employing algebraic techniques like expansion and factoring showcases understanding and facilitates simplification.
• Performing logical operations like isolating variables or dividing both sides by a common term moves the proof forward.
• Concluding with simplification or re-expression proves the identity, reinforcing the concept's validity.

Proofs not only establish truth but also build a framework for solving similar problems. Practicing this method sharpens analytical skills, making it easier to handle trigonometric identities and complex mathematical reasoning.