Problem 96 - A Graphical Approach to Precalculus with Limits

Question

Verify that each equation is an identity by using any of the identities introduced in the first three sections of this chapter. $$\frac{2}{1+\cos x}-\tan ^{2} \frac{x}{2}=1$$

Step-by-Step Solution

Answer

Yes, the equation is an identity.

Step 1: Simplify the Left Side of the Equation

We start with the left side of the equation: \( \frac{2}{1+\cos x} - \tan^2 \frac{x}{2} \). First, simplify \( \frac{2}{1+\cos x} \). We know from trigonometric identities that \( 1+\cos x = 2 \cos^2 \frac{x}{2} \), so \( \frac{2}{1+\cos x} = \frac{2}{2 \cos^2 \frac{x}{2}} = \sec^2 \frac{x}{2} \).

Step 2: Use Trigonometric Identity for \( \tan^2 \frac{x}{2} \)

From trigonometric identities, we know that \( \sec^2 \theta - 1 = \tan^2 \theta \). In our case, substitute \( \theta = \frac{x}{2} \), so \( \sec^2 \frac{x}{2} - \tan^2 \frac{x}{2} = 1 \).

Step 3: Conclusion of Identity Verification

By using the identity \( \sec^2 \theta - \tan^2 \theta = 1 \), we have shown that \( \sec^2 \frac{x}{2} - \tan^2 \frac{x}{2} = 1 \). Therefore, the original equation \( \frac{2}{1+\cos x} - \tan^2 \frac{x}{2} = 1 \) is indeed an identity.

Key Concepts

Identity Verification

Identity verification in trigonometry involves proving that two different expressions are equivalent for all values within the range of the variable. In the given exercise, we are tasked with confirming the identity:\( \frac{2}{1+\cos x} - \tan^2 \frac{x}{2}=1 \). To verify this identity, a deep understanding of trigonometric properties is necessary.

We start by transforming each side of the equation using known identities to either reach the same expression or a simple form like 1. In this case, substituting equivalent forms based on trigonometric identities steps the calculation forward. For identity verification, it is crucial to manipulate one or both sides of the equation to show that they are equal.

Remember, the purpose of verifying an identity is to confirm its validity universally, not just under specific circumstances. This strengthens problem-solving by reinforcing familiarity with how different expressions can simplify to the same value.

Trigonometric Simplification

Trigonometric simplification is the process of making a trigonometric expression more manageable or recognizable by using relationships between functions. For the given problem, we used this process to simplify the left side of the equation \( \frac{2}{1+\cos x} - \tan^2 \frac{x}{2} \).

Simplification started with \( \frac{2}{1+\cos x} \). By employing the identity \( 1+\cos x = 2 \cos^2 \frac{x}{2} \), we simplified this to \( \sec^2 \frac{x}{2} \). It's important to notice how recognizing and applying identities can lead to considerable simplification.

Effective simplification often requires a good grasp of basic and derived trigonometric identities, such as Pythagorean identities and double-angle formulas. Practice in these transformations improves both efficiency and accuracy in solving trigonometric equations.

Half-Angle Identities

Half-angle identities are invaluable tools in trigonometry, used to express trigonometric functions of half an angle in terms of the square roots of expressions with functions of the full angle. They are particularly useful for simplifying or transforming expressions that involve a half-angle, as seen in our problem with \( \tan^2 \frac{x}{2} \).

The half-angle identity for cosine, for example, is \( \cos \frac{x}{2} = \pm \sqrt{\frac{1+\cos x}{2}} \). In our exercise, understanding that \( 1+\cos x = 2 \cos^2 \frac{x}{2} \) allowed us to simplify expressions with confidence and clarity.

Half-angle identities are part of the standard toolkit for trigonometric transformations and simplifications. Mastery of these identities supports better skills in identity verification and solving a wide range of trigonometric problems efficiently.