Problem 13 - Algebra 2

Question

Verify that each of the following is an identity. $$ \frac{1-2 \cos ^{2} \theta}{\sin \theta \cos \theta}=\tan \theta-\cot \theta $$

Step-by-Step Solution

Answer

The identity is verified: \( \tan \theta - \cot \theta \).

Step 1: Simplify the Left-Hand Side (LHS)

Start by simplifying the left-hand side expression: \( \frac{1-2 \cos^2 \theta}{\sin \theta \cos \theta} \). Recognize that \( 1 - 2 \cos^2 \theta \) can be rewritten using the double angle identity as \(-\cos 2\theta\). Thus, the LHS becomes \( \frac{-\cos 2\theta}{\sin \theta \cos \theta} \).

Step 2: Express LHS in Terms of Sine and Cosine

Now, rewrite the expression in step 1: \( \frac{-\cos 2\theta}{\sin \theta \cos \theta} = \frac{-(\cos^2 \theta - \sin^2 \theta)}{\sin \theta \cos \theta} \).

Step 3: Use the Double Angle Identity

Using \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \), simplify the numerator. Hence, the expression becomes: \( - \frac{\cos^2 \theta - \sin^2 \theta}{\sin \theta \cos \theta} \).

Step 4: Simplify the Expression Further

The current form is \( - \frac{\cos^2 \theta - \sin^2 \theta}{\sin \theta \cos \theta} = \frac{\sin^2 \theta - \cos^2 \theta}{\sin \theta \cos \theta} \).

Step 5: Separate into Two Fractions

Split the fraction: \( \frac{\sin^2 \theta}{\sin \theta \cos \theta} - \frac{\cos^2 \theta}{\sin \theta \cos \theta} \) simplifies to \( \frac{\sin \theta}{\cos \theta} - \frac{\cos \theta}{\sin \theta} \).

Step 6: Simplify to Match Right-Hand Side (RHS)

Recognize that \( \frac{\sin \theta}{\cos \theta} = \tan \theta \) and \( \frac{\cos \theta}{\sin \theta} = \cot \theta \). Thus, the expression becomes \( \tan \theta - \cot \theta \). This matches the right-hand side of the identity.

Key Concepts

Double Angle Identity

The concept of the double angle identity in trigonometry is crucial for simplifying complex expressions involving angles. One of the commonly used double angle identities is for cosine, which states that \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \). This identity can be incredibly useful:

• It simplifies the expression \( 1 - 2\cos^2 \theta \) as \( -\cos 2\theta \).
• This allows us to manipulate and reduce expressions by transforming them into more manageable forms involving sine and cosine.

Understanding double angle identities helps in transforming trigonometric expressions into simpler ones, which can then be easily compared or verified against other forms. In the context of verifying trigonometric identities, recognizing these transformations is key to connecting different forms of the same identity.

Simplifying Expressions

Simplifying expressions is a critical skill in mathematics, especially in the field of trigonometry. The process involves reducing complex trigonometric expressions into simpler forms using known identities and algebraic manipulations.

• In the given problem, the expression \( \frac{1-2 \cos^2 \theta}{\sin \theta \cos \theta} \) is simplified using the double angle identity for cosine.
• This reduces the problem to \( \frac{-\cos 2\theta}{\sin \theta \cos \theta} \), which can be broken down further into simpler fractions.

The art of simplification helps in making complicated expressions more understandable and easier to work with. By simplifying, one can quickly identify patterns or equivalent expressions, such as transforming \( \frac{\sin^2 \theta}{\sin \theta \cos \theta} - \frac{\cos^2 \theta}{\sin \theta \cos \theta} \) into simpler trigonometric functions like tangent and cotangent.

Verifying Identities

Verifying trigonometric identities involves demonstrating that two sides of an equation are equivalent by using algebraic and trigonometric manipulations. This requires a solid understanding of trigonometric identities and simplification techniques.

• In our example, we start with the expression on the left-hand side and simplify it to eventually match the right-hand side, \( \tan \theta - \cot \theta \).
• By applying identities such as the double angle identity and simplifying each part of the expression, we're able to verify that both sides of the given equation match perfectly.

Successfully verifying identities helps build confidence in understanding trigonometric relationships and enhances problem-solving skills in mathematics. It's a strategic approach that trains one to think critically about different methods to equate two seemingly different expressions.