Problem 9 - Algebra 2

Question

Simplify each trigonometric expression. $$ \tan \theta \cot \theta $$

Step-by-Step Solution

Answer

The simplified expression of \( \tan\theta \cot\theta \) is 1.

Step 1: Understanding Tangent and Cotangent

Firstly, let's recall that \( \tan\theta = \frac{\sin\theta}{\cos\theta} \) and \( \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta} \). These are basic trigonometric identities.

Step 2: Substitute the Identities into the Expression

Now, substitute these identities into the given expression. This will give us \( \tan\theta \cot\theta = \frac{\sin\theta}{\cos\theta} * \frac{\cos\theta}{\sin\theta} \).

Step 3: Simplify

Finally simplify the expression. When we multiply these two fractions, \( \frac{\sin\theta}{\cos\theta} * \frac{\cos\theta}{\sin\theta} \), you'll notice that sin and cos cancel out in the numerator and denominator, which will leave you with 1.

Key Concepts

Tangent and Cotangent Identities

Understanding tangent and cotangent is crucial for simplifying trigonometric expressions. The tangent function, denoted as \( \tan \theta \), can be expressed in terms of sine and cosine functions: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This relationship highlights how tangent is essentially the ratio of the opposite side to the adjacent side in a right-angled triangle, which are represented by the sine and cosine of that angle, respectively.

The cotangent function is the reciprocal of the tangent. Written as \( \cot \theta \), it is calculated as \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \). By understanding these identities, you can more easily interchange or substitute the trigonometric functions in problems, which is often a strategy employed in simplification and other trigonometric manipulations.

Both these functions play a significant role in simplifying trigonometric expressions and prove useful in transforming equations into formats that are easier to work with.

Trigonometric Simplification

Trigonometric simplification involves expressing complex trigonometric expressions in their simplest forms. This is often done by using known identities such as those for tangent and cotangent, and substituting them into the expression is a key step in simplification. For example, if you start with the expression \( \tan \theta \cot \theta \), you'll first substitute the identities to obtain \( \tan \theta \cot \theta = \frac{\sin \theta}{\cos \theta} \times \frac{\cos \theta}{\sin \theta} \).

Upon substitution, you can see that the sine and cosine terms in the numerators and denominators can be canceled. Simplifying means eliminating terms that appear both in the numerator and the denominator, like \( \sin \theta \) and \( \cos \theta \) in this instance. This results in simplifying the expression just to \( 1 \). This process demonstrates how seemingly complicated ratios reduce down to simple numerical results. Simplicity not only brings clarity but also aids in further mathematical processing and interpretation.

Algebraic Manipulation

Algebraic manipulation is a mathematical technique used to change the form of an equation or expression. In trigonometry, it is often used together with trigonometric identities to simplify expressions like \( \tan \theta \cot \theta \).

In the step-by-step solution provided in the original exercise, algebraic manipulation started when the identities of tangent and cotangent were substituted into the expression. This allowed the terms to multiply and cancel each other out due to their reciprocal nature. It’s important to understand that multiplication can allow for terms in the numerator and denominator to cancel out when they are identical, resulting in a simplified equation or expression.

This type of rearrangement requires a strong grasp of basic algebraic principles, including multiplication, division, and the concept of reciprocal numbers. Mastering these skills is essential for tackling more advanced mathematical problems and for successfully simplifying various expressions in calculus and algebra.