Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
1 + cot(-θ)/cot(-θ)

Verified step by step guidance

1. Recall the definition of cotangent in terms of sine and cosine: \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\). Therefore, \(\cot(-\theta) = \frac{\cos(-\theta)}{\sin(-\theta)}\).
2. Use the even-odd properties of cosine and sine: \(\cos(-\theta) = \cos(\theta)\) (cosine is even) and \(\sin(-\theta) = -\sin(\theta)\) (sine is odd). Substitute these into the expression for \(\cot(-\theta)\) to get \(\cot(-\theta) = \frac{\cos(\theta)}{-\sin(\theta)} = -\frac{\cos(\theta)}{\sin(\theta)}\).
3. Rewrite the original expression \(1 + \frac{\cot(-\theta)}{\cot(-\theta)}\) by substituting the simplified form of \(\cot(-\theta)\): \(1 + \frac{-\frac{\cos(\theta)}{\sin(\theta)}}{-\frac{\cos(\theta)}{\sin(\theta)}}\).
4. Simplify the fraction \(\frac{-\frac{\cos(\theta)}{\sin(\theta)}}{-\frac{\cos(\theta)}{\sin(\theta)}}\). Since numerator and denominator are the same, this fraction simplifies to 1.
5. Combine the terms: \$1 + 1 = 2$. The expression is now simplified with no quotients and only sine and cosine functions of \(\theta\).

Key Concepts

Trigonometric Functions and Their Definitions

Understanding the basic trigonometric functions—sine, cosine, and cotangent—is essential. Cotangent is defined as the ratio of cosine to sine (cot θ = cos θ / sin θ). Expressing all functions in terms of sine and cosine helps simplify complex expressions.

Even-Odd Properties of Trigonometric Functions

Knowing how trigonometric functions behave with negative angles is crucial. Sine is an odd function (sin(-θ) = -sin θ), while cosine is even (cos(-θ) = cos θ). Cotangent, being a quotient of cosine and sine, is an odd function (cot(-θ) = -cot θ). This helps simplify expressions involving negative angles.

Simplification of Trigonometric Expressions

Simplifying expressions involves rewriting quotients as products or sums without fractions, and combining like terms. The goal is to express the entire expression using only sine and cosine functions of θ, eliminating quotients and negative angles for clarity and ease of further manipulation.