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The double angle formulas are an essential tool in trigonometry. They help you transform expressions involving trigonometric functions into forms that are easier to work with.
These formulas involve functions like sine, cosine, and tangent when the angle is doubled.
For the sine function, the double angle formula is:

• \( \sin 2\theta = 2 \sin \theta \cos \theta \)

This formula shows us how to express \( \sin 2\theta \) in terms of \( \sin \theta \) and \( \cos \theta \). This is particularly useful for simplifying products like the one in the exercise.
Understanding these formulas can make solving trigonometric identities much simpler. In practice, applying the formula means finding \( \theta \) and then expressing the double angle equation, which we can use to simplify and solve expressions.

The sine function, one of the basic trigonometric functions, is incredibly important in mathematics. It is often abbreviated as \( \sin \). This function represents the ratio of the opposite side to the hypotenuse in a right triangle.
The sine function is periodic with a period of \( 2\pi \), which means that its values repeat every full rotation.
Key values for the sine function at common angles include:

• \( \sin 0 = 0 \)
• \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)
• \( \sin \frac{\pi}{2} = 1 \)
• \( \sin \pi = 0 \)

These values help you quickly simplify expressions. In the exercise, knowing that \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \) allows for a straightforward simplification of the final expression.

Trigonometric simplification involves rewriting expressions in a simpler form, often using known identities or values. This can make it easier to solve equations or understand relationships between angles.
In the exercise, simplifying the product \( \sin \frac{\pi}{8} \cos \frac{\pi}{8} \) is done by applying the double angle formula and recognizing familiar trigonometric values.
Key strategies for simplification include:

• Identifying which trigonometric identity to use, like double angle formulas.
• Substituting known values for angles, such as \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).
• Simplifying fractions and expressions by performing basic arithmetic operations.

Understanding these strategies helps in tackling more complex trigonometric problems and achieving a clear and concise result in simplification tasks.