Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.
cot(-θ)/sec(-θ)

Verified step by step guidance

1. Recall the definitions and properties of the trigonometric functions involved: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \). Also, remember the even-odd properties: \( \cos(-\theta) = \cos \theta \) (even function) and \( \sin(-\theta) = -\sin \theta \) (odd function).
2. Rewrite \( \cot(-\theta) \) using the definition of cotangent and the odd-even properties: \( \cot(-\theta) = \frac{\cos(-\theta)}{\sin(-\theta)} = \frac{\cos \theta}{-\sin \theta} = -\frac{\cos \theta}{\sin \theta} \).
3. Rewrite \( \sec(-\theta) \) using the definition of secant and the even property of cosine: \( \sec(-\theta) = \frac{1}{\cos(-\theta)} = \frac{1}{\cos \theta} \).
4. Form the quotient \( \frac{\cot(-\theta)}{\sec(-\theta)} \) by substituting the expressions from steps 2 and 3: \( \frac{-\frac{\cos \theta}{\sin \theta}}{\frac{1}{\cos \theta}} \).
5. Simplify the complex fraction by multiplying numerator and denominator appropriately to eliminate the quotient: \( -\frac{\cos \theta}{\sin \theta} \times \frac{\cos \theta}{1} = -\frac{\cos^2 \theta}{\sin \theta} \). This expression is now in terms of \( \sin \theta \) and \( \cos \theta \) only, with no quotients involving other trigonometric functions.

Key Concepts

Even and Odd Trigonometric Functions

Understanding the parity of trigonometric functions is essential for simplifying expressions with negative angles. Sine is an odd function, meaning sin(-θ) = -sin(θ), while cosine is even, so cos(-θ) = cos(θ). Secant and cotangent inherit parity from cosine and sine respectively, affecting how negative angles are handled.

Trigonometric Identities for Cotangent and Secant

Cotangent and secant can be expressed in terms of sine and cosine: cot(θ) = cos(θ)/sin(θ) and sec(θ) = 1/cos(θ). Rewriting these functions helps to convert the given expression into a form involving only sine and cosine, facilitating simplification without quotients.

Simplification of Trigonometric Expressions

After rewriting the expression in terms of sine and cosine, algebraic manipulation is used to eliminate quotients and combine terms. This involves multiplying numerator and denominator appropriately and applying fundamental identities to achieve a simplified expression solely in sin(θ) and cos(θ).