Solution: Use identity $\sin(2z) = 2\sin z \cos z$. So:

["Mastering Trigonometric Identities: How $\sin(2z) = 2\sin z \cos z$ Simplifies Complex Calculations", "Understanding trigonometric identities is essential for students, engineers, and researchers tackling problems in calculus, physics, and signal processing. One of the most powerful and frequently used identities is:", "$$\n\sin(2z) = 2\sin z \cos z\n$$", "This double-angle identity not only simplifies trigonometric equations but also enhances efficiency in solving integrals, derivatives, and waveform analysis. In this article, we explore how leveraging this identity transforms complex calculations into straightforward solutions.", "---", "### What Is the Identity $\sin(2z) = 2\sin z \cos z$?", "The identity $\sin(2z) = 2\sin z \cos z$ expresses the sine of double an angle in terms of the sine and cosine of the original angle. It is derived from the standard sine addition formula:", "$$\n\sin(a + b) = \sin a \cos b + \cos a \sin b\n$$", "Setting $a = b = z$, we get:", "$$\n\sin(2z) = \sin(z + z) = \sin z \cos z + \cos z \sin z = 2\sin z \cos z\n$$", "This elegant relationship allows quick computation and simplification when working with trigonometric functions involving doubled angles.", "---", "### Why Use This Identity?", "1. Simplifies Calculations\n Instead of expanding $\sin(2z)$ using the full addition formula, applying $\sin(2z) = 2\sin z \cos z$ often saves time and reduces error—especially useful in rapid problem-solving or exams.", "2. Facilitates Integration\n When integrating expressions like $\sin(2z)\cos(z)$, recognizing the double-angle identity allows rewriting the integrand into a product of sines and cosines that is easier to integrate using standard techniques.", "3. Aids in Derivative Computation\n Differentiating expressions involving $\sin(2z)$ becomes direct when using the identity: derivative of $2\sin z \cos z$ is simpler than differentiating the original form.", "---", "### Real-World Applications", "- Physics & Engineering:\n In wave interference and oscillatory motion, $\sin(2z)$ frequently arises. Using this identity clarifies phase shifts and amplitude modulation.", "- Signal Processing:\n Modulated signals often use double-angle identities to analyze frequency components and cross-correlations.", "- Calculus & Differential Equations:\n Many differential equations involving trigonometric terms resolve more cleanly through this identity.", "---", "### How to Apply It Step-by-Step", "Example: Simplify $\sin(2z)\cos(z)$", "Step 1: Apply the identity $\sin(2z) = 2\sin z \cos z$\n$$\n\sin(2z)\cos z = (2\sin z \cos z)\cos z\n$$", "Step 2: Simplify using algebra\n$$\n= 2\sin z \cos^2 z\n$$", "This form is often more useful than expanding further or using power-reduction formulas.", "---", "### Practical Tips for Use", "- Memorize the identity—quick recall speeds up problem-solving.\n- Alternate forms when convenient: You may convert $\sin(2z)$ to $2\sin z \cos z$ or $\cos(2z)$ depending on context.\n- Combine with other identities like $\cos^2 z = \frac{1 + \cos(2z)}{2}$ for advanced integrals.", "---", "### Conclusion", "The identity $\sin(2z) = 2\sin z \cos z$ is far more than a formula—it's a foundational tool that streamlines trigonometric problem-solving across multiple disciplines. By mastering its use, students and professionals can tackle complex equations with confidence and precision.", "Next time you encounter a problem involving double angles, remember: the simplicity of $\sin(2z)$ awaits—just triple the r tsp. $\boxed{\sin(2z) = 2\sin z \cos z}$ and simplify your work!", "---", "Keywords: sin(2z), identity, trigonometric identity, double angle formula, simplify trigonometry, calculus, physics applications, signal processing, integration, differentiation, mathematics education."]