Problem 17 - Precalculus: Functions and Graphs

Question

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: \(\frac{\tan^2 x}{\sec x+1} = \frac{1-\cos x}{\cos x}\).

Step 1: Rewrite Trigonometric Functions Using Identities

Start by expressing both sides of the identity in terms of sine and cosine. We know that:\[\tan x = \frac{\sin x}{\cos x}, \quad \sec x = \frac{1}{\cos x}\]So, rewrite \(\tan^2 x\) as:\[\tan^2 x = \frac{\sin^2 x}{\cos^2 x}\]And \(\sec x + 1\) as:\[\sec x + 1 = \frac{1}{\cos x} + 1 = \frac{1 + \cos x}{\cos x}\]

Step 2: Substitute and Simplify the Left Side

Substitute the rewritten expressions into the left side of the original equation:\[\frac{\tan^{2}x}{\sec x + 1} = \frac{\frac{\sin^2 x}{\cos^2 x}}{\frac{1 + \cos x}{\cos x}}\]Now, simplify the fraction by multiplying by the reciprocal of the denominator:\[\frac{\sin^2 x}{\cos^2 x} \times \frac{\cos x}{1 + \cos x} = \frac{\sin^2 x}{\cos x (1 + \cos x)}\]

Step 3: Simplify the Right Side

Now consider the right side of the original identity:\[\frac{1 - \cos x}{\cos x}\]This is already simplified as it only needs to be expressed in terms of division. There is no further simplification needed here.

Step 4: Verify the Equality

Now compare the simplified forms from step 2 and step 3. We have: \[\frac{\sin^2 x}{\cos x (1 + \cos x)}\]And from step 3: \[\frac{1 - \cos x}{\cos x}\]Recall the Pythagorean identity: \[\sin^2 x = 1 - \cos^2 x\]Notice:\[\sin^2 x = (1 - \cos x)(1 + \cos x)\]Thus, \[\frac{(1 - \cos x)(1 + \cos x)}{\cos x (1 + \cos x)} = \frac{1 - \cos x}{\cos x}\]Both sides of the identity equate, therefore it is verified.

Key Concepts

Sine and Cosine Identities

Trigonometric identities are formulas involving trigonometric functions, such as sine and cosine. These identities can simplify complex expressions and verify equivalencies. In this exercise, we start by expressing tangent and secant in terms of sine and cosine, because they are fundamental trigonometric functions:

• The tangent function is \( \tan x = \frac{\sin x}{\cos x} \).
• The secant function is \( \sec x = \frac{1}{\cos x} \).

Understanding these relations is crucial because they help us rewrite more complex trigonometric expressions into simpler forms involving just sine and cosine. By converting into sine and cosine, we have a better chance of revealing hidden simplifications or verifying identities. Always remember that manipulating expressions using identities might include rewriting fractions or using the reciprocal of a function.

Pythagorean Identity

One of the most important identities in trigonometry is the Pythagorean identity, which states:\[ \sin^2 x + \cos^2 x = 1 \]This equation is fundamental because it relates the square of sine and cosine functions. In our verification problem, understanding how this identity helps break down expressions is key.

• For example, \( \sin^2 x \) can be rewritten as \( 1 - \cos^2 x \).
• Expanding \( (1 - \cos x)(1 + \cos x) \) reveals \( 1 - \cos^2 x \), simplifying identification of equivalent expressions.

Using the Pythagorean identity allows you to substitute one part of an expression with another, leading to further simplifications. During solutions, recognize parts of expressions that can be transformed using this identity.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is an essential skill in solving identities and equations. It involves rewriting the expression in a simpler or more familiar form without changing its value. Here’s how to approach simplification:

• Convert complex trigonometric functions into sine and cosine.
• Factor expressions to identify common patterns or terms.
• Simplify fractions by multiplying by the reciprocal.
• Use trigonometric identities to eliminate or reduce terms.

In our identity verification, these steps allowed us to transform the left side of the equation to match the right side. Breaking down \( \frac{\tan^2 x}{\sec x + 1} \) into simpler terms revealed a straightforward match with \( \frac{1 - \cos x}{\cos x} \). Always check each algebraic manipulation, ensuring you’re maintaining equivalence between expressions at every step.