To solve the equation
\[ \frac{\sec x}{1 + \cot^2 x} \]
we can first rewrite the components in terms of sine and cosine.
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Recall the trigonometric identities:
- \(\sec x = \frac{1}{\cos x}\)
- \(\cot^2 x = \frac{\cos^2 x}{\sin^2 x}\)
- Using the Pythagorean identity, \(1 + \cot^2 x = \frac{\sin^2 x + \cos^2 x}{\sin^2 x} = \frac{1}{\sin^2 x}\).
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Plug these into the equation: \[ \frac{\sec x}{1 + \cot^2 x} = \frac{\frac{1}{\cos x}}{\frac{1}{\sin^2 x}} = \frac{\sin^2 x}{\cos x}. \]
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Rewriting \(\frac{\sin^2 x}{\cos x}\) yields: \[ \frac{\sin^2 x}{\cos x} = \tan x \cdot \sin x. \]
Thus, the expression that completes the equation is:
\(\tan x \sin x\).
The correct response among the options given is:
tan x sin x.