Problem 11 - Algebra 2

Question

Verify that each of the following is an identity. $$ \sin \theta \sec \theta \cot \theta=1 $$

Step-by-Step Solution

Answer

The identity \( \sin \theta \sec \theta \cot \theta = 1 \) is verified by simplification.

Step 1: Substitute Trigonometric Identities

Let's substitute the trigonometric identities into the given expression. We have:- \( \sec \theta = \frac{1}{\cos \theta} \)- \( \cot \theta = \frac{1}{\tan \theta} \)Now substitute these into the expression \( \sin \theta \sec \theta \cot \theta \):\[ \sin \theta \cdot \frac{1}{\cos \theta} \cdot \frac{1}{\tan \theta} \]

Step 2: Simplify Using Basic Trigonometric Ratios

Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Therefore, \( \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \).Substitute \( \frac{1}{\tan \theta} \):\[ \sin \theta \cdot \frac{1}{\cos \theta} \cdot \frac{\cos \theta}{\sin \theta} \]

Step 3: Cancel Out Terms

In the expression \( \sin \theta \cdot \frac{1}{\cos \theta} \cdot \frac{\cos \theta}{\sin \theta} \), both \( \sin \theta \) and \( \cos \theta \) terms can be canceled. This simplifies to:\[ \frac{\sin \theta}{\sin \theta} \cdot \frac{\cos \theta}{\cos \theta} = 1 \]

Step 4: Verify the Identity

Since simplifying the expression gives us 1, this confirms that the original trigonometric expression is indeed an identity.

Key Concepts

Trigonometric Ratios

Trigonometric ratios are fundamental in understanding relationships within a right-angled triangle. These ratios include sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)). The reciprocal functions are also key elements:

• Secant (\( \sec \theta = \frac{1}{\cos \theta} \))
• Cosecant (\( \csc \theta = \frac{1}{\sin \theta} \))
• Cotangent (\( \cot \theta = \frac{1}{\tan \theta} \))

Understanding these trigonometric identities is crucial because they are often used in simplifying expressions and verifying identities. For instance, in the given exercise, the identities for secant and cotangent were used to transform the expression into a more manageable form. This makes it easier to perform subsequent mathematical operations. Internalizing these fundamental trigonometric ratios allows students to tackle more complex problems understanding how these basic ratios combine and transform.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves rewriting complex expressions using basic trigonometric identities. It requires both substitution of identities and reduction of terms. In the given problem, the expression \( \sin \theta \cdot \sec \theta \cdot \cot \theta \) is simplified by substituting identities:

• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

This simplifies the expression to \( \sin \theta \cdot \frac{1}{\cos \theta} \cdot \frac{\cos \theta}{\sin \theta} \), using the relationship \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) for further simplification. The process continues with canceling like terms, in this case, \( \sin \theta \) and \( \cos \theta \), reducing the expression to 1.
Successfully simplifying expressions like this can make solving equations much simpler, as it reduces the problem to basic arithmetic.

Verifying Identities

Verifying trigonometric identities is an essential skill in trigonometry that helps ensure the equality of two expressions. The goal is to show that one side of an equation can be transformed into the other using known identities. In our exercise, verifying the identity means showing that \( \sin \theta \cdot \sec \theta \cdot \cot \theta = 1 \). Through substitution and simplification, we confirmed this by

• Replacing complex parts of the expression with simpler trigonometric identities
• Simplifying and canceling terms

The step-by-step approach turns the original expression into 1, verifying it is indeed an identity. Developing the ability to verify identities is key in solving trigonometric equations reliably and efficiently. It builds a strong foundation for understanding trigonometric properties, functions, and their interrelationships.