Simplify each expression.
√[(1 + cos 76°)/2]

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Recognize that the expression inside the square root, \(\frac{1 + \cos 76^\circ}{2}\), matches the form of the half-angle identity for cosine: \(\cos^2 \left( \frac{\theta}{2} \right) = \frac{1 + \cos \theta}{2}\).

Identify the angle \(\theta\) in the expression as \(76^\circ\), so the half-angle is \(\frac{76^\circ}{2} = 38^\circ\).

Rewrite the expression inside the square root using the half-angle identity: \(\frac{1 + \cos 76^\circ}{2} = \cos^2 38^\circ\).

Since the original expression is \(\sqrt{\frac{1 + \cos 76^\circ}{2}}\), substitute the equivalent expression to get \(\sqrt{\cos^2 38^\circ}\).

Simplify the square root of the square to \(|\cos 38^\circ|\). Because \(38^\circ\) is in the first quadrant where cosine is positive, this simplifies to \(\cos 38^\circ\).

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Half-Angle Identity for Cosine

The half-angle identity expresses the cosine of an angle in terms of the cosine of half that angle. Specifically, (1 + cos θ)/2 equals cos²(θ/2). This identity helps simplify expressions involving cosines of angles by relating them to squares of cosines of half-angles.

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When simplifying square roots of squared trigonometric functions, the result is the absolute value of the original function. For example, √(cos² α) = |cos α|, ensuring the output is non-negative, which is important when simplifying expressions involving square roots.

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Understanding how to evaluate trigonometric functions at given angles, such as 38° (half of 76°), is essential. This involves either using a calculator or known values to find the cosine or sine of specific angles, enabling the simplification of expressions involving numerical angle measures.

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