In Exercises 63–68, find the exact value of each expression. Do not use a calculator.csc 37° sec 53° - tan 53° cot 37°

Verified step by step guidance

1. Recognize that \( \csc 37^\circ = \frac{1}{\sin 37^\circ} \) and \( \sec 53^\circ = \frac{1}{\cos 53^\circ} \).
2. Notice that \( \tan 53^\circ = \frac{\sin 53^\circ}{\cos 53^\circ} \) and \( \cot 37^\circ = \frac{1}{\tan 37^\circ} = \frac{\cos 37^\circ}{\sin 37^\circ} \).
3. Use the complementary angle identities: \( \sin 37^\circ = \cos 53^\circ \) and \( \cos 37^\circ = \sin 53^\circ \).
4. Substitute these identities into the expression: \( \csc 37^\circ \sec 53^\circ - \tan 53^\circ \cot 37^\circ \).
5. Simplify the expression using the identities: \( \frac{1}{\sin 37^\circ} \cdot \frac{1}{\cos 53^\circ} - \frac{\sin 53^\circ}{\cos 53^\circ} \cdot \frac{\cos 37^\circ}{\sin 37^\circ} \).

Key Concepts

Cosecant and Secant Functions

Cosecant (csc) and secant (sec) are reciprocal trigonometric functions. Cosecant is the reciprocal of sine, defined as csc(θ) = 1/sin(θ), while secant is the reciprocal of cosine, defined as sec(θ) = 1/cos(θ). Understanding these functions is essential for evaluating expressions involving angles, particularly when calculating exact values without a calculator.

Tangent and Cotangent Functions

Tangent (tan) and cotangent (cot) are also fundamental trigonometric functions. Tangent is defined as tan(θ) = sin(θ)/cos(θ), while cotangent is its reciprocal, cot(θ) = cos(θ)/sin(θ). These functions are crucial for simplifying expressions that involve angles, especially when combined with their reciprocal relationships.

Angle Relationships in Trigonometry

In trigonometry, certain angle relationships can simplify calculations. For example, the angles 37° and 53° are complementary, meaning that sin(37°) = cos(53°) and vice versa. Recognizing these relationships allows for the substitution of trigonometric functions, facilitating the evaluation of expressions involving multiple angles.