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. Recall the complementary angle relationships: since 37° and 53° add up to 90°, we have \( \sin 37^\circ = \cos 53^\circ \) and \( \cos 37^\circ = \sin 53^\circ \). This will help simplify the trigonometric expressions.
2. Rewrite each trigonometric function in terms of sine and cosine: \( \csc 37^\circ = \frac{1}{\sin 37^\circ} \), \( \sec 53^\circ = \frac{1}{\cos 53^\circ} \), \( \tan 53^\circ = \frac{\sin 53^\circ}{\cos 53^\circ} \), and \( \cot 37^\circ = \frac{\cos 37^\circ}{\sin 37^\circ} \).
3. Substitute these into the expression: \( \csc 37^\circ \sec 53^\circ - \tan 53^\circ \cot 37^\circ = \frac{1}{\sin 37^\circ} \times \frac{1}{\cos 53^\circ} - \frac{\sin 53^\circ}{\cos 53^\circ} \times \frac{\cos 37^\circ}{\sin 37^\circ} \).
4. Use the complementary angle identities to replace \( \cos 53^\circ \) with \( \sin 37^\circ \) and \( \sin 53^\circ \) with \( \cos 37^\circ \), simplifying the expression to terms involving only \( \sin 37^\circ \) and \( \cos 37^\circ \).
5. Simplify the resulting expression by combining fractions and canceling common terms to find the exact value without using a calculator.

Key Concepts

Reciprocal Trigonometric Functions

Cosecant (csc) and secant (sec) are the reciprocals of sine and cosine, respectively. Specifically, csc θ = 1/sin θ and sec θ = 1/cos θ. Understanding these relationships helps simplify expressions involving these functions without a calculator.

Complementary Angle Relationships

Angles like 37° and 53° are complementary because they add up to 90°. Trigonometric functions of complementary angles are related, for example, sin(37°) = cos(53°) and tan(37°) = cot(53°). This property is useful for rewriting and simplifying expressions.

Simplification of Trigonometric Expressions

Simplifying expressions involves substituting reciprocal and complementary angle identities and combining terms. Recognizing patterns and using fundamental identities allows finding exact values without a calculator, which is essential for solving the given problem.