Perform each transformation. See Example 2.
Write sec x in terms of sin x.

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Recall the definition of secant in terms of cosine: \(\sec x = \frac{1}{\cos x}\).

Use the Pythagorean identity relating sine and cosine: \(\sin^2 x + \cos^2 x = 1\).

Solve the identity for \(\cos x\): \(\cos x = \pm \sqrt{1 - \sin^2 x}\).

Substitute \(\cos x\) into the secant expression: \(\sec x = \frac{1}{\pm \sqrt{1 - \sin^2 x}}\).

Note that the sign depends on the quadrant of \(x\), so the expression for \(\sec x\) in terms of \(\sin x\) is \(\sec x = \pm \frac{1}{\sqrt{1 - \sin^2 x}}\).

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Reciprocal Trigonometric Functions

The secant function, sec x, is defined as the reciprocal of the cosine function, i.e., sec x = 1/cos x. Understanding this relationship is essential to rewrite sec x in terms of other trigonometric functions.

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The fundamental identity sin²x + cos²x = 1 allows expressing cosine in terms of sine: cos x = ±√(1 - sin²x). This identity is crucial for converting sec x into an expression involving sin x.

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When expressing sec x in terms of sin x, the sign of cos x (and thus sec x) depends on the quadrant of x. Recognizing the domain restrictions ensures the correct sign is chosen for the square root in the transformation.

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