4sin2phicos2phi

The expression 4sin²(φ)cos²(φ) is a trigonometric identity that can be simplified. This expression is equivalent to sin²(2φ).

To see this, we can use the double angle identity for sine, which states that sin(2φ) = 2sin(φ)cos(φ). Squaring both sides of this identity gives us sin²(2φ) = (2sin(φ)cos(φ))². Expanding the right side, we get sin²(2φ) = 4sin²(φ)cos²(φ). Therefore, the original expression is a direct result of squaring the double angle formula for sine.

This identity is useful in various areas of mathematics, including calculus and physics, where simplifying trigonometric expressions can make calculations more manageable. For example, in integration, recognizing this identity can lead to a simpler integral to evaluate. It also appears in the study of wave phenomena and oscillations, where amplitudes and intensities are often described using trigonometric functions. The simplification allows for a more direct relationship between the angle φ and the resulting squared sine value.