Problem 66 - A Graphical Approach to Precalculus with Limits

Question

Verify that each equation is an identity. $$\cos x=\frac{1-\tan ^{2} \frac{x}{2}}{1+\tan ^{2} \frac{x}{2}}$$

Step-by-Step Solution

Answer

The equation is an identity because it follows from cosine half-angle identities.

Step 1: Simplify the Right Side

We need to verify the identity: \( \cos x = \frac{1-\tan^2 \frac{x}{2}}{1+\tan^2 \frac{x}{2}} \). Start by recognizing the identity for cosine in terms of half-angle tangent: \( \cos x = \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} \) for \( \theta = \frac{x}{2} \). So the right side is already in a form using this identity.

Step 2: Use Half-Angle Identities

Recall the half-angle identity for tangent: \( \tan \frac{x}{2} = \frac{1 - \cos x}{\sin x} \). In this equation, it's expressed that the terms \( \tan \frac{x}{2} \) can produce \( \cos x \) when arranged and squared properly, indicating that \( \cos x = \frac{1 - \tan^2 \frac{x}{2}}{1 + \tan^2 \frac{x}{2}} \) is derived from these identities.

Step 3: Transform Right Side

To rigorously verify, assume \( u = \tan \frac{x}{2} \), then we have from trigonometric identities \( \cos x = \frac{1-u^2}{1+u^2} \), where this comes from reconciling the half-angle substitutions back into full-angle trigonometric definitions.

Step 4: Conclusion

Since through transformations consistent with trigonometric identities, \( \cos x = \frac{1-u^2}{1+u^2} \) holds for \( u = \tan \frac{x}{2} \), the given equation is an identity.

Key Concepts

Half-Angle Identities

Half-angle identities are essential tools in trigonometry, helpful for relating the trigonometric functions of half-angles to the values of the original angle. Such identities allow us to simplify complex expressions and verify various trigonometric identities. These are helpful when solving equations or when transformations between functions are needed.

When working with half-angles, you transform a full-angle problem into a more manageable form using these identities. For the tangent half-angle identity, expressed as \( \tan \frac{x}{2} = \frac{1 - \cos x}{\sin x} \), it connects the tangent of a half-angle to the sine and cosine of the full angle. This expression helps bridge different functions and simplifies verification of identities involving tan, cos, and sin at angles divided by two.

By understanding and applying half-angle identities, you can easily navigate complex trigonometric identities, simplifying or even transforming them into different forms. They emphasize the deep relationship between the different trigonometric functions.

Cosine Function

The cosine function, denoted as \( \cos \), is one of the primary trigonometric functions. It gives the x-coordinate of a point on the unit circle as an angle \( x \) is swept from the positive x-axis.

The cosine of an angle is a fundamental part of many trigonometric identities and can be represented in numerous forms using identities such as the double angle, sum and difference, and, importantly, half-angle identities. For instance, using half-angle transformations, the cosine function's value can be rewritten in a different form involving the tangent of half the angle: \( \cos x = \frac{1 - \tan^2 \frac{x}{2}}{1 + \tan^2 \frac{x}{2}} \). This form is particularly useful when angles need to be expressed in terms of other basic trigonometric functions.

• Cosine is periodic, with a period of \( 2\pi \).
• It is an even function, meaning \( \cos(-x) = \cos(x) \).
• Cosine values range from -1 to 1.

Understanding the cosine function's properties and its relation to other identities helps in solving and proving trigonometric equations.

Tangent Function

The tangent function, denoted as \( \tan \), is a fundamental trigonometric function that represents the ratio of the sine and cosine of an angle \( x \). For any angle, it can be expressed as \( \tan x = \frac{\sin x}{\cos x} \). This makes tangent particularly useful in scenarios where both sine and cosine need to be considered together.

The tangent function is closely tied to various trigonometric identities, especially as part of angle transformations such as the half-angle identity: \( \tan \frac{x}{2} = \frac{1 - \cos x}{\sin x} \). This identity helps transform expressions involving full angles into forms predominantly involving half-angles, which can simplify solving complex trigonometric identities and integrals.

• Tangent is periodic, with a period of \( \pi \).
• It is an odd function, meaning \( \tan(-x) = -\tan(x) \).
• Tangent does not have a maximum or minimum value, as it extends to infinity.

By exploring the properties and transformations related to the tangent function, you gain a deeper understanding of its applications in trigonometry and its role in constructing and verifying identities.