Problem 13 - Precalculus

Question

Use the fundamental identities to fully simplify the expression. $$\frac{1+\tan ^{2} \theta}{\csc ^{2} \theta}+\sin ^{2} \theta+\frac{1}{\sec ^{2} \theta}$$

Step-by-Step Solution

Answer

The simplified expression is \(\sec^2 \theta\).

Step 1: Apply Trigonometric Identity for Tangent and Secant

Recall the Pythagorean identity: \(1 + \tan^2 \theta = \sec^2 \theta\). Substitute this identity into the expression: \(\frac{1 + \tan^2 \theta}{\csc^2 \theta} + \sin^2 \theta + \frac{1}{\sec^2 \theta}\) becomes \(\frac{\sec^2 \theta}{\csc^2 \theta} + \sin^2 \theta + \frac{1}{\sec^2 \theta}\).

Step 2: Convert Secant and Cosecant to Sine and Cosine

Convert \(\sec^2 \theta\) and \(\csc^2 \theta\) to their corresponding forms using sine and cosine: \(\sec^2 \theta = \frac{1}{\cos^2 \theta}\) and \(\csc^2 \theta = \frac{1}{\sin^2 \theta}\). Thus, the expression becomes \(\frac{\frac{1}{\cos^2 \theta}}{\frac{1}{\sin^2 \theta}} + \sin^2 \theta + \frac{1}{\frac{1}{\cos^2 \theta}}\).

Step 3: Simplify the Fraction

Perform the division in the fraction: \(\frac{\frac{1}{\cos^2 \theta}}{\frac{1}{\sin^2 \theta}} = \frac{\sin^2 \theta}{\cos^2 \theta}\). The expression now is \(\frac{\sin^2 \theta}{\cos^2 \theta} + \sin^2 \theta + \cos^2 \theta\).

Step 4: Simplify the Expression Further Using Pythagorean Identity

Recognize that \(\sin^2 \theta + \cos^2 \theta = 1\) using the Pythagorean identity. Substitute this into the expression: \(\frac{\sin^2 \theta}{\cos^2 \theta} + 1\).

Step 5: Recognize and Express as a Single Fraction

Rewrite the expression as a single fraction: \( \frac{\sin^2 \theta}{\cos^2 \theta} + 1 = \frac{\sin^2 \theta + \cos^2 \theta}{\cos^2 \theta} \). Since \( \sin^2 \theta + \cos^2 \theta = 1 \), the expression simplifies to \(\frac{1}{\cos^2 \theta}\), which equals \(\sec^2 \theta\).

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry that connects the squares of sine and cosine. It is expressed as \[ \sin^2 \theta + \cos^2 \theta = 1. \] Consider this identity as the trigonometric equivalent of the Pythagorean Theorem. It essentially tells us that the squares of the sine and cosine of the same angle will always add up to one. For example, if you take an angle like \( \theta \), and calculate \( \sin^2 \theta \) and \( \cos^2 \theta \), their sum will be 1, no matter the specific angle.
The identity becomes particularly handy when simplifying expressions, as seen in the original exercise. For instance, it allows us to conveniently replace terms like \( \sin^2 \theta + \cos^2 \theta \) with 1, reducing the complexity of expressions. This identity is key in transforming expressions to simpler forms and solving equations effectively.

Sine and Cosine Relationships

Sine and cosine are not just fundamental trigonometric functions; they are interconnected with other functions such as tangent, secant, cosecant, and cotangent. Understanding these relationships can make simplifying expressions much easier.

• Tangent: Defined as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). When you apply a Pythagorean identity, you find \( 1 + \tan^2 \theta = \sec^2 \theta \), a handy equation in many simplifications.
• Secant: Secant relates to cosine as \( \sec \theta = \frac{1}{\cos \theta} \). It is the reciprocal function of cosine, meaning secant values help express cosine in fraction terms.
• Cosecant: Similarly, \( \csc \theta = \frac{1}{\sin \theta} \). This reciprocal relationship simplifies transformations between functions.

By switching between these relationships, you can transform and simplify trigonometric expressions. For instance, in the given problem, secant and cosecant are expressed in terms of sine and cosine to make the expression easier to handle.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves reducing complex expressions to their simplest form using identities and relationships. This requires recognizing patterns and identities, then applying them strategically.

• First, identify any known identities, like \( 1 + \tan^2 \theta = \sec^2 \theta \) or the basic Pythagorean identity.
• Replace compound fractions with equivalent simpler forms using reciprocal functions such as secant, cosecant, and tangent expressed in sine and cosine.
• Combine like terms and rewrite the expression using a common denominator if possible.

In the example problem, we simplify the given expression step by step by substituting identities and converting functions. Operations like division and combining fractions lead to a final simplified expression of \( \sec^2 \theta \), demonstrating how the strategic use of identities can streamline trigonometric calculations.