Problem 69 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

Yes, \( \cos 2x = \frac{1-\tan^2 x}{1+\tan^2 x} \) is an identity.

Step 1: Recalling the Double Angle Formula for Cosine

The double angle formula for cosine is given by \( \cos 2x = \cos^2 x - \sin^2 x \). This can also be expressed in terms of tangent as \( \cos 2x = \frac{1 - \tan^2 x}{1 + \tan^2 x} \). Our goal in this problem is to verify this identity by manipulating the right-hand side of the equation.

Step 2: Substituting Trigonometric Identities

We know the identities \( \sin x = \tan x \cdot \cos x \) and \( 1 + \tan^2 x = \sec^2 x \). Therefore, we substitute \( \sec^2 x \) into the denominator. The expression becomes \[ \frac{1 - \tan^2 x}{\sec^2 x}\].

Step 3: Simplification Using Reciprocal Identity

Recall that \( \sec x = \frac{1}{\cos x} \). Thus, \( \sec^2 x = \frac{1}{\cos^2 x} \), which simplifies our expression to \( (1 - \tan^2 x) \times \cos^2 x \).

Step 4: Using Pythagorean Identity

The Pythagorean identity is \( 1 - \tan^2 x = \cos^2 x - \sin^2 x \). Substitute this into the expression to get \( (\cos^2 x - \sin^2 x) \times \cos^2 x \) which simplifies to \( \cos^2 x - \sin^2 x \).

Step 5: Final Verification

The expression \( \cos^2 x - \sin^2 x \) is identical to the double angle formula \( \cos 2x \), confirming that the original equation is indeed an identity.

Key Concepts

Double Angle Formula

The double angle formulas are among the essentials in trigonometry. They allow you to express trigonometric functions of double angles in terms of single angles. The double angle formula for cosine, in particular, is \( \cos 2x = \cos^2 x - \sin^2 x \). This equation can also appear in a version that uses tangent:

• \( \cos 2x = \frac{1 - \tan^2 x}{1 + \tan^2 x} \)

This alternative form is crucial when dealing with equations involving tangent. By using these formulas, we can transform complex expressions into simpler ones that are easier to analyze or verify, as it was demonstrated in the exercise.
To make these connections clear, remember that using \( \cos 2x = \cos^2 x - \sin^2 x \) often involves turning expressions of sine or cosine into terms of tangent, using identities to navigate between these forms.

Tangent and Secant Identities

Understanding tangent and secant identities can significantly simplify trigonometric equations. Let's begin with the relationship between tangent and secant. The identity \( 1 + \tan^2 x = \sec^2 x \) is particularly useful when transforming expressions. This identity reflects how tangent and secant are related through the Pythagorean theorem applied to trigonometric functions.
Additionally, the reciprocal identity \( \sec x = \frac{1}{\cos x} \) helps in converting secant into a more manageable expression involving cosine. This is exactly what was used in the solution to the exercise. By substituting \( \sec^2 x \) with \( \frac{1}{\cos^2 x} \), the equation becomes simpler and more straightforward to handle, thus paving the way for further simplifications. Overall, mastering these identities helps tremendously in verifying and proving trigonometric identities.

Pythagorean Identity

The Pythagorean identity is one of the fundamental building blocks of trigonometry. It is given by:

• \( 1 - \tan^2 x = \cos^2 x - \sin^2 x \)

This identity connects sine, cosine, and tangent in a useful way that’s often needed in various calculations, including our original problem. By substituting it into trigonometric expressions, simplifications can be achieved that allow further manipulations, leading to verification or transformation of an identity.
In the exercise solution, this identity was the key to converting between tangent and the traditional double angle formula for cosine, illustrating its application in trigonometric transformations. Mastery of this identity allows for smoother navigation through complex trigonometric expressions, aiding in unraveling them down to their core components.