Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Verify the identity. $$ \frac{\tan ^{2} x}{\sec x+1}=\frac{1-\cos x}{\cos x} $$

Step-by-Step Solution

Answer

The identity is verified; both sides equal \( \frac{1 - \cos x}{\cos x} \).

Step 1: Recall Trigonometric Identities

To simplify the given identity \( \frac{\tan^2 x}{\sec x + 1} = \frac{1 - \cos x}{\cos x} \), we should recall that \( \tan x = \frac{\sin x}{\cos x} \) and \( \sec x = \frac{1}{\cos x} \). These identities will help us transform one side of the equation into the other.

Step 2: Simplify the Left-Hand Side (LHS)

Substitute \( \tan x = \frac{\sin x}{\cos x} \) into \( \tan^2 x \) to get \( \tan^2 x = \frac{\sin^2 x}{\cos^2 x} \). The LHS becomes \( \frac{\sin^2 x}{\cos^2 x \cdot (\sec x + 1)} \). Substitute \( \sec x = \frac{1}{\cos x} \) to get \( \sec x + 1 = \frac{1}{\cos x} + 1 = \frac{1 + \cos x}{\cos x} \). Therefore, the LHS becomes \( \frac{\sin^2 x \cos x}{(1 + \cos x) \cos^2 x} = \frac{\sin^2 x}{\cos x (1 + \cos x)} \).

Step 3: Rearrange the LHS

Notice that \( \sin^2 x = 1 - \cos^2 x \). Substitute this identity into the LHS: \( \frac{1 - \cos^2 x}{\cos x (1 + \cos x)} \). Factor the numerator \( 1 - \cos^2 x = (1 - \cos x)(1 + \cos x) \). Thus the LHS becomes \( \frac{(1 - \cos x)(1 + \cos x)}{\cos x (1 + \cos x)} \).

Step 4: Cancel Common Factors

The term \( (1 + \cos x) \) appears in both the numerator and the denominator, so they can be cancelled out. This leaves us with \( \frac{1 - \cos x}{\cos x} \), which is the same as the right-hand side (RHS).

Step 5: Verify

Since the left-hand side \( \frac{1 - \cos x}{\cos x} \) is equivalent to the right-hand side, the identity is verified: \( \frac{\tan^2 x}{\sec x + 1} = \frac{1 - \cos x}{\cos x} \).

Key Concepts

Trigonometric Simplification

Trigonometric simplification involves reducing complex trigonometric expressions into simpler forms. This process is crucial for solving equations or verifying identities. Easily overlooked details in the simplification process can make equations appear much more complex than necessary. In trigonometric functions, simplification often involves using basic trigonometric identities, such as:

• \( \tan x = \frac{\sin x}{\cos x} \)
• \( \sec x = \frac{1}{\cos x} \)

By substituting these identities and systematically reducing parts of the expression, we can focus on getting rid of fractions and unnecessary terms. For example, in the exercise given, the left-hand side (LHS) expression \( \frac{\tan^2 x}{\sec x + 1} \) was simplified using these key identities. The replacement of \( \tan^2 x \) with \( \frac{\sin^2 x}{\cos^2 x} \), and then proceeding to incorporate the secant identity, shows the step-by-step simplification process, making it clear and easy to follow. This practice not only makes the problem less intimidating but also illuminates relationships between different trigonometric functions, which is essential for verifying identities and solving complex trigonometric equations.

Angle Identities

Angle identities are essential tools when working with trigonometric functions. They help us rewrite expressions involving trigonometric angles in different ways. Common angle identities include sum and difference identities, double-angle identities, and half-angle identities. Understanding these can be particularly helpful when simplifying expressions or equations.
In our exercise, the identity \( \sin^2 x = 1 - \cos^2 x \) played a vital role in simplifying the expression \( \frac{\sin^2 x}{\cos x (1 + \cos x)} \) to \( \frac{(1 - \cos x)(1 + \cos x)}{\cos x (1 + \cos x)} \). This ultimately facilitated the canceling out of redundant terms in the numerator and the denominator.
Such angle identities are powerful because they allow us to manipulate trigonometric expressions in ways that reveal hidden equivalences. They often seem like small steps but are crucial in transforming an entire side of an equation during the verification process, as seen in this exercise.

Verifying Identities

Verifying identities involves proving that two distinct trigonometric expressions are equivalent to each other. The main goal is to transform both sides of the identity into the same form, thus showing their equivalence. This is typically done by selecting one side of the equation and using trigonometric identities to manipulate it until it matches the other side.
In this particular exercise, we demonstrated verification by simplifying the left-hand side (LHS) \( \frac{\tan^2 x}{\sec x + 1} \) into the right-hand side (RHS) \( \frac{1 - \cos x}{\cos x} \). By methodically reducing the LHS with substitutive identities like \( \tan^2 x = \frac{\sin^2 x}{\cos^2 x} \) and \( \sec x = \frac{1}{\cos x} \), each step followed logically to reveal that the two sides of the equation are indeed equal.

• Identify applicable identities
• Simplify each expression using these identities
• Cancel out common terms to reveal equivalence

This practical approach helps in gaining an intuitive understanding of how trigonometric expressions are interrelated, making it less daunting to tackle other similar problems. Enhanced familiarity with these steps is key to mastering this concept.