Problem 154 - Precalculus

Question

For the following exercises, prove the identities. $$ \cos (2 \alpha)=\frac{1-\tan ^{2} \alpha}{1+\tan ^{2} \alpha} $$

Step-by-Step Solution

Answer

The identity is proven using the double angle formula for cosine and expressing in terms of \( \tan^2(\alpha) \).

Step 1: Recall trigonometric identities

We should recall the double angle formula for cosine: \( \cos(2\alpha) = \cos^2(\alpha) - \sin^2(\alpha) \). Also, we derive identities for tangent: \( \tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)} \).

Step 2: Express sine and cosine in terms of tangent

Using \( \tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)} \), we can express \( \sin(\alpha) = \tan(\alpha) \cdot \cos(\alpha) \). Consequently, \( \sin^2(\alpha) = \tan^2(\alpha) \cdot \cos^2(\alpha) \). Recall that \( \cos^2(\alpha) + \sin^2(\alpha) = 1 \).

Step 3: Substitute expressions into cosine double angle formula

Substitute \( \sin^2(\alpha) = \tan^2(\alpha) \cdot \cos^2(\alpha) \) into \( \cos(2\alpha) = \cos^2(\alpha) - \sin^2(\alpha) \). It becomes \( \cos(2\alpha) = \cos^2(\alpha) - \tan^2(\alpha) \cdot \cos^2(\alpha) \).

Step 4: Factor out \( \cos^2(\alpha) \)

Factor out \( \cos^2(\alpha) \): \( \cos(2\alpha) = \cos^2(\alpha)(1 - \tan^2(\alpha)) \).

Step 5: Express everything in terms of \( \tan^2(\alpha) \)

Use the identity \( \cos^2(\alpha) = \frac{1}{1 + \tan^2(\alpha)} \). Substitute this to get \( \cos(2\alpha) = \frac{1}{1 + \tan^2(\alpha)} \cdot (1 - \tan^2(\alpha)) \).

Step 6: Simplify the expression

Simplify \( \cos(2\alpha) = \frac{1 - \tan^2(\alpha)}{1 + \tan^2(\alpha)} \). This matches the given identity, completing the proof.

Key Concepts

Double Angle Formula

The double angle formulae are essential tools in trigonometry, used to simplify complex calculations involving angles. One of the core double angle identities is for cosine:

• \( \cos(2\alpha) = \cos^2(\alpha) - \sin^2(\alpha) \)
• Alternatively, it can also be expressed as \( \cos(2\alpha) = 2\cos^2(\alpha) - 1 \)
• Or, \( \cos(2\alpha) = 1 - 2\sin^2(\alpha) \)

These forms demonstrate how we can express double angles in terms of single angle trigonometric functions.
By substituting different expressions of the sine and cosine functions, we can tailor these identities to solve specific problems. Understanding these formulae offers a foundation for tackling many trigonometric simplifications or proofs.
In exercises, always remember: first, identify which form of the double angle formula will simplify the task at hand most efficiently.

Tangent Identity

The tangent function bridges the sine and cosine functions, defined as the ratio:

• \( \tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)} \)
• This identity allows us to express sine and cosine in terms of tangent because each can derive from the other.

In the context of the cosine double angle formula, expressing \( \sin(\alpha) \) and \( \cos(\alpha) \) in terms of \( \tan(\alpha) \) helps us achieve a simpler form.
For instance, suppose \( \sin(\alpha) = \tan(\alpha) \cdot \cos(\alpha) \). Consequently, \( \sin^2(\alpha) = \tan^2(\alpha) \cdot \cos^2(\alpha) \). It links the relationship between these three functions.
The comprehension of this tangent identity is central to transforming expressions with sine and cosine into those made up of tangent, simplifying the problem-solving process when tackling trigonometric identities.

Cosine Identity

The cosine identity is an aspect of trigonometry that provides relationships among angles and enriches the understanding of trigonometric properties. In proving identities, the following Pythagorean identity should always be at the forefront:

• \( \cos^2(\alpha) + \sin^2(\alpha) = 1 \)

This identity relates sine and cosine to establish their complementary roles in the unit circle. By rearranging this identity, you can express \( \cos^2(\alpha) \) as \( \cos^2(\alpha) = 1 - \sin^2(\alpha) \) or in terms of tangent, \( \cos^2(\alpha) = \frac{1}{1 + \tan^2(\alpha)} \).
It becomes particularly handy when proving identities or manipulating expressions like in the substitution steps shown in exercises.
This understanding is critical not just in proofs, but also in comprehending broader trigonometric concepts that involve algebraic manipulations among sine, cosine, and tangent functions.