Problem 14 - Algebra and Trigonometry

Question

Verify each identity. \(\frac{\cos \theta \sec \theta}{\cot \theta}=\tan \theta\)

Step-by-Step Solution

Answer

The identity \(\frac{\cos \theta \sec \theta}{\cot \theta}=\tan \theta\) is verified as correct at \(\tan \theta = \tan \theta \)}

Step 1: Identify Trigonometric Identities

First, get a grip on identities involved. Here, \(\cos \theta\), \(\sec \theta\), and \(\cot \theta\) are involved. Knowing that \(\sec \theta = 1/ \cos \theta\) and \(\cot \theta = 1/ \tan \theta\) will be valuable.

Step 2: Start Substitute the Reciprocal Identities

Now, substitute the reciprocal identities into the given equation. It then becomes \(\frac{\cos \theta \cdot (1/ \cos \theta)}{1/ \tan \theta}\). Simplify this by cancelling out \(\cos \theta\) in the numerator and it becomes \(\frac{1}{1/ \tan \theta}\).

Step 3: Multiply by Reciprocal

Now, simplify \(\frac{1}{1/ \tan \theta}\) by multiplying by its own reciprocal which is simply \(\tan \theta\). This is done as follows: \(\frac{1}{1/ \tan \theta} \cdot \tan \theta = \tan \theta\) and hence verified the initial identity

Key Concepts

Reciprocal Identities

In trigonometry, understanding reciprocal identities is crucial for simplifying expressions and verifying identities. Reciprocal identities express the relationship between certain trigonometric functions and their reciprocals. For example, the secant function \(\sec \theta\) is the reciprocal of the cosine function \(\cos \theta\), which means:

• \(\sec \theta = \frac{1}{\cos \theta}\)
• \(\cos \theta = \frac{1}{\sec \theta}\)
• Similarly, the cotangent function \(\cot \theta\) is the reciprocal of the tangent function \(\tan \theta\): \(\cot \theta = \frac{1}{\tan \theta}\)

Using these identities, we can transform trigonometric expressions into forms that are easier to manage. This method often simplifies solving equations or verifying more complex identities.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves using identities and algebraic manipulation to transform expressions into their simplest forms. In the example above, you start by substituting reciprocal identities into the equation:
\(\frac{\cos \theta \sec \theta}{\cot \theta}\).
Replace \(\sec \theta\) with \(\frac{1}{\cos \theta}\) and \(\cot \theta\) with \(\frac{1}{\tan \theta}\), which results in:
\(\frac{\cos \theta \cdot \frac{1}{\cos \theta}}{\frac{1}{\tan \theta}}\).
This expression simplifies to \(\frac{1}{\frac{1}{\tan \theta}}\). By recognizing that dividing by a fraction is equivalent to multiplying by its reciprocal, the expression further reduces to \(\tan \theta\).

• This process highlights the power of reciprocal identities in simplification tasks.

It makes complex expressions more manageable, paving the way for easier verification of identities.

Verifying Identities

Verifying trigonometric identities is like solving a puzzle. You start with one side of the equation and manipulate it to show that it equals the other side. In this process, your knowledge of trigonometric identities, like reciprocal identities, plays a vital role.
Using the original identity \(\frac{\cos \theta \sec \theta}{\cot \theta} = \tan \theta\), you manipulate one side to match the other. After simplifying as shown above, you find that both sides equal \(\tan \theta\). This verifies the identity.

• This verification is important because it confirms that the identity holds true for all values of \(\theta\) within the allowable range.
• It reinforces the consistency of trigonometric rules and relationships.

Effective verification demands a solid understanding of different identities and strategic manipulation using algebraic techniques.