Simplify each expression.
sin 158.2°/(1 + cos 158.2°)

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Recognize that the expression \( \frac{\sin 158.2^\circ}{1 + \cos 158.2^\circ} \) can be simplified using trigonometric identities related to half-angle formulas or sum-to-product formulas.

Recall the identity for tangent of half an angle: \( \tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta} \). This matches the form of the given expression.

Apply the identity by setting \( \theta = 158.2^\circ \), so the expression simplifies to \( \tan \frac{158.2^\circ}{2} \).

Calculate the half-angle: \( \frac{158.2^\circ}{2} = 79.1^\circ \), so the expression becomes \( \tan 79.1^\circ \).

Thus, the simplified form of the original expression is \( \tan 79.1^\circ \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all angle values. Key identities like the Pythagorean identity and angle sum/difference formulas help simplify expressions by rewriting them in more manageable forms.

Half-Angle and Sum/Difference Formulas

Half-angle and sum/difference formulas allow the expression of trigonometric functions of certain angles in terms of other angles. These formulas are useful for simplifying expressions involving angles that are not standard, such as 158.2°, by relating them to more familiar angles.

Simplification of Rational Trigonometric Expressions

Simplifying rational expressions involving sine and cosine often involves factoring, using identities, or converting the expression into a single trigonometric function. Recognizing patterns like sin θ / (1 + cos θ) can lead to simpler forms such as tangent of half the angle.

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