Problem 55 - Algebra and Trigonometry

Question

Verify each identity. \(\left(\tan ^{2} \theta+1\right)\left(\cos ^{2} \theta+1\right)=\tan ^{2} \theta+2\)

Step-by-Step Solution

Answer

Yes, the identity \((\tan^2\theta+1)(\cos^2\theta+1) = \tan^2\theta + 2\) is correct.

Step 1: Expand the left-hand side

Begin by expanding the left-hand side of the equation: \((\tan^2\theta + 1)(\cos^2\theta + 1) = \tan^2\theta \cos^2\theta + \tan^2\theta + \cos^2\theta + 1\)

Step 2: Substitute for \(\tan^2\theta\)

Remember that \(\tan^2\theta + 1 = \sec^2\theta\). So, substitute \(\tan^2\theta\) with \(\sec^2\theta - 1\). Doing so gives us \((\sec^2\theta - 1)\cos^2\theta + (\sec^2\theta - 1) + \cos^2\theta + 1 = \sec^2\theta \cos^2\theta - \cos^2\theta + \sec^2\theta - 1 + \cos^2\theta + 1\)

Step 3: Simplify the equation

Simplify this down gives us: \(\sec^2\theta \cos^2\theta + \sec^2\theta\). Remember that \(\sec\theta = 1/\cos\theta\) so \(\sec^2\theta \cos^2\theta = 1\). This simplifies to \(1 + \sec^2\theta = 1 + \tan^2\theta +1\), which simplifies further to \(\tan^2\theta + 2\), the RHS.

Key Concepts

Tan Squared Theta Identity

Understanding the identity involving \textbf{tan squared theta} is pivotal in trigonometry. This identity is a part of the Pythagorean identities and is derived from the basic relationship of a right triangle's sides. The identity states that \(\tan^2\theta + 1 = \text{sec}^2\theta\). In simple terms, if you square the tangent of an angle and add one, you get the square of the secant of that angle. The tangent squared identity can be thought of as a transformation that allows us to switch between tan and sec with ease.

For example, in the provided exercise, the given equation has \(\tan^2\theta\) on both sides. By using the identity mentioned above, we can replace \(\tan^2\theta + 1\) with \(\text{sec}^2\theta\), which simplifies the problem and moves us closer to verification. Understanding this identity and its applications is fundamental to simplifying trigonometric expressions involving tangents.

Trigonometric Identity Simplification

The process of \textbf{trigonometric identity simplification} is akin to unraveling a knot. The goal is to take a complex expression and reduce it to its simplest form without changing its value. Simplification often involves recognizing and applying fundamental trigonometric identities such as Pythagorean identities, quotient identities, and reciprocal identities.

To simplify trigonometric identities effectively:

• Look for patterns that match known trigonometric identities.
• Replace complex expressions with simpler equivalents if possible.
• Factor and cancel terms where applicable.
• Combine like terms to reach the simplest form.

In the exercise at hand, by applying the tan squared theta identity, we substitute to make further simplifications. Also, understanding that \(\text{sec}\theta\) is the reciprocal of \(\text{cos}\theta\) allows us to convert between these functions and ultimately simplify the complex expression to a simple one that verifies the identity.

Secant Squared Theta Identity

The \textbf{secant squared theta identity} is intrinsically related to the tangent squared theta identity. As mentioned, \(\text{sec}^2\theta - 1 = \tan^2\theta\). The secant function, which is the reciprocal of the cosine, often appears less frequently in calculations than sine or cosine but plays a critical role in simplifying expressions that involve the tangent. Importantly, \(\text{sec}\theta = \frac{1}{\text{cos}\theta}\), so \(\text{sec}^2\theta = \frac{1}{\text{cos}^2\theta}\).

This relationship turns multiplication involving secants and cosines into much simpler forms. For instance, in the solution steps, \(\text{sec}^2\theta \text{cos}^2\theta\) becomes 1, fantastic evidence of how identities can dramatically simplify expressions. The secant squared theta identity is typically used within the context of calculus as well when working with integrals and derivatives pertaining to trigonometric functions.