Question

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.(sin θ - cos θ) (csc θ + sec θ)

Accepted Answer

Start by rewriting the given expression $$(\sin 	heta - \cos 	heta)(\csc 	heta + \sec 	heta) in $$terms of sine and cosine. Recall that $$\csc 	heta = \frac{1}{\sin 	heta}$$ and $$\sec 	heta = \frac{1}{\cos 	heta}$$, so substitute these into the expression. The expression becomes $$(\sin 	heta - \cos 	heta) \left( \frac{1}{\sin 	heta} + \frac{1}{\cos 	heta} ight). $$Next, combine the terms inside the parentheses over a common denominator $$\sin 	heta \cos 	heta. $$Rewrite the sum inside the parentheses as $$\frac{\cos 	heta}{\sin 	heta \cos 	heta} + \frac{\sin 	heta}{\sin 	heta \cos 	heta} = \frac{\cos 	heta + \sin 	heta}{\sin 	heta \cos 	heta}. $$Now the expression is $$(\sin 	heta - \cos 	heta) \cdot \frac{\cos 	heta + \sin 	heta}{\sin 	heta \cos 	heta}. $$Multiply the numerators: $$(\sin 	heta - \cos 	heta)(\cos 	heta + \sin 	heta). $$Recognize this as a product of two binomials and expand it using the distributive property (FOIL method). After expanding, simplify the numerator by combining like terms. Then, write the entire expression as a single fraction with denominator $$\sin 	heta \cos 	heta. $$Finally, look for any trigonometric identities or algebraic simplifications to eliminate quotients and express the result purely in terms of $$\sin 	heta$$ and $$\cos 	heta$$ without fractions.

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.
(sin θ - cos θ) (csc θ + sec θ)

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Key Concepts

Trigonometric Functions and Their Reciprocal Identities

Algebraic Manipulation of Trigonometric Expressions

Avoiding Quotients in Trigonometric Simplification