Question

Simplify. $$\sin \theta(\csc \theta+\cot \theta)$$

Short answer

\(1 + \cos \theta\)

Step-by-step solution

Identify the trigonometric identities

First, recognize that \(\csc \theta = \frac{1}{\sin \theta}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). These identities will be used to simplify the given expression.

Rewrite the expression using identities

Rewrite \(\sin \theta(\csc \theta+\cot \theta)\) by substituting the identities, to get \(\sin \theta\left(\frac{1}{\sin \theta} + \frac{\cos \theta}{\sin \theta}\right)\).

Simplify the expression

Distribute \(\sin \theta\) to both terms inside the parenthesis to obtain \(\sin \theta \cdot \frac{1}{\sin \theta} + \sin \theta \cdot \frac{\cos \theta}{\sin \theta}\). This simplifies to \(1 + \cos \theta\), as \(\sin \theta / \sin \theta = 1\).

Key concepts

Simplifying Trigonometric Expressions

Understanding how to simplify trigonometric expressions is a vital skill in geometry and calculus, as it allows for the easier evaluation and solution of problems. Simplification often involves applying trigonometric identities—equations that are true for all values of the involved variables.

The key to simplifying an expression, like in the given exercise \(\sin \theta(\csc \theta+\cot \theta)\), starts with recognizing and rewriting using trigonometric identities. By replacing \(\csc \theta\) with \(\frac{1}{\sin \theta}\) and \(\cot \theta\) with \(\frac{\cos \theta}{\sin \theta}\), the expression becomes easier to work with.

Once the identities are substituted in, algebraic simplification comes into play. Terms that feature the same trigonometric function in both numerator and denominator, such as \(\sin \theta / \sin \theta\), can be cancelled out to 1. Additionally, watch out for opportunities to combine like terms and simplify complex fractions. By systematically applying these strategies, even the most complicated trigonometric expressions can be broken down into simpler forms.

Cosecant (csc)

The cosecant function, often denoted \(\csc\), is one of the basic trigonometric functions. It is actually the reciprocal of the sine function. This means that for any angle \(\theta\), \(\csc \theta = \frac{1}{\sin \theta}\). It's essential to recognize that \(\csc\) is undefined whenever \(\sin\) is 0—this occurs at angle measures of 0°, 180°, and their coterminal angles.

In trigonometry, the cosecant function arises in various contexts such as in solving triangles, graphing trigonometric functions, and in calculus. Understanding how to manipulate and simplify expressions involving \(\csc\) is critical, especially when converting between different trigonometric forms or simplifying complex expressions. Substituting \(\csc\) with its equivalent in terms of sine can often reveal simpler paths to a solution, as seen in the given exercise.

Cotangent (cot)

The cotangent function, written as \(\cot\), is another trigonometric ratio that students encounter. It is defined as the reciprocal of the tangent function, or the ratio of the adjacent side to the opposite side in a right-angled triangle. Algebraically, we express it as \(\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\).

In practice, \(\cot\) is useful for finding angles or side lengths in right triangles where the lengths of two sides are known. Additionally, it's often seen in calculus, particularly in integration and differentiation of trigonometric functions. Simplifying expressions with \(\cot\), as demonstrated in the exercise, involves recognizing its definition in terms of sine and cosine, which can unlock further simplification through cancellation or rearrangement of terms.