Perform each indicated operation and simplify the result so that there are no quotients.
sec x/csc x + csc x/sec x

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Recall the definitions of the secant and cosecant functions in terms of sine and cosine: \(\sec x = \frac{1}{\cos x}\) and \(\csc x = \frac{1}{\sin x}\).

Rewrite each quotient in the expression \(\frac{\sec x}{\csc x} + \frac{\csc x}{\sec x}\) by substituting the definitions: \(\frac{\frac{1}{\cos x}}{\frac{1}{\sin x}} + \frac{\frac{1}{\sin x}}{\frac{1}{\cos x}}\).

Simplify each complex fraction by multiplying numerator and denominator appropriately: \(\frac{1}{\cos x} \times \frac{\sin x}{1} + \frac{1}{\sin x} \times \frac{\cos x}{1}\).

This simplifies to \(\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}\), which are simpler trigonometric expressions without quotients of sec and csc.

To combine and simplify further, find a common denominator \(\sin x \cos x\) and write the expression as \(\frac{\sin^2 x}{\sin x \cos x} + \frac{\cos^2 x}{\sin x \cos x}\), then combine the numerators.

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Reciprocal Trigonometric Functions

Secant (sec x) and cosecant (csc x) are reciprocal functions of cosine and sine, respectively. Specifically, sec x = 1/cos x and csc x = 1/sin x. Understanding these relationships helps in rewriting expressions to simplify or eliminate quotients.

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Complex fractions involve fractions within fractions. To simplify, rewrite all terms with common denominators or multiply numerator and denominator by the least common denominator to eliminate nested fractions, making the expression easier to handle.

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Using fundamental identities like sin²x + cos²x = 1 allows for rewriting and simplifying expressions. Recognizing opportunities to combine terms or convert to sine and cosine helps reduce the expression to a simpler form without quotients.

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