Question

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

Accepted Answer

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. Rewrite each trigonometric function in terms of sine and cosine: $$ \csc 37^\circ = \frac{1}{\sin 37^\circ} $$, $$ \sec 53^\circ = \frac{1}{\cos 53^\circ} $$, $$ 	an 53^\circ = \frac{\sin 53^\circ}{\cos 53^\circ} $$, and $$ \cot 37^\circ = \frac{\cos 37^\circ}{\sin 37^\circ} . $$Substitute these into the expression: $$ \csc 37^\circ \sec 53^\circ - 	an 53^\circ \cot 37^\circ = \frac{1}{\sin 37^\circ} 	imes \frac{1}{\cos 53^\circ} - \frac{\sin 53^\circ}{\cos 53^\circ} 	imes \frac{\cos 37^\circ}{\sin 37^\circ} . $$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 $$ \cos 37^\circ . $$Simplify the resulting expression by combining fractions and canceling common terms to find the exact value without using a calculator.

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

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Key Concepts

Reciprocal Trigonometric Functions

Complementary Angle Relationships

Simplification of Trigonometric Expressions