Using the Addition Formulas Use the addition formulas to derive the identities 34. \( \sin \left(x-\frac{\pi}{2}\right)=-\cos x \)

To derive the identity \( \sin \left(x - \frac{\pi}{2}\right) = -\cos x \), we can employ the sine subtraction formula: \[ \sin(a - b) = \sin a \cos b - \cos a \sin b \] In our case, let \( a = x \) and \( b = \frac{\pi}{2} \). Plugging these values into the formula gives us: \[ \sin\left(x - \frac{\pi}{2}\right) = \sin x \cos\left(\frac{\pi}{2}\right) - \cos x \sin\left(\frac{\pi}{2}\right) \] Since \( \cos\left(\frac{\pi}{2}\right) = 0 \) and \( \sin\left(\frac{\pi}{2}\right) = 1 \), we simplify to: \[ \sin\left(x - \frac{\pi}{2}\right) = \sin x \cdot 0 - \cos x \cdot 1 = -\cos x \] Thus, the identity is confirmed! Now, isn't it fascinating how these little angles can lead to whole new geometric truths? It's like uncovering secret portals in the world of trigonometry! When you flip those angles around, they transform parts of the wave, unveiling relationships you might overlook at first glance. Dive deeper, and you might just find more cool transformations! And speaking of angles, the relationship between sine and cosine can be a lot of fun! Next time you're playing around with triangles or wave functions, remember how critical these identities can be. They not only help simplify expressions but also aid in solving complex equations in calculus and physics. So keep them in your toolkit; you never know when they'll come in handy!