Consider the expression: \( \frac{1-\cos ^{2} P}{4 \cos \left(90^{\circ}+P\right)} \) 1. Simplify the expressian to a single trigonometric term. Answer: \( \frac{-\sin P}{4} \) \( \frac{\cos P}{4} \) \( \frac{\sin P}{4} \) \( \frac{4 \sin P}{} \)

The expression \( \frac{1-\cos^{2} P}{4 \cos(90^{\circ}+P)} \) can be simplified step by step. First, recall that \( 1 - \cos^2 P = \sin^2 P \). So, the numerator becomes \( \sin^2 P \). Next, using the fact that \( \cos(90^{\circ}+P) = -\sin P \), we can rewrite the denominator: \[ \cos(90^{\circ}+P) = -\sin P \] Now, substituting these results into the expression gives: \[ \frac{\sin^2 P}{4(-\sin P)} = -\frac{\sin P}{4} \] So the simplified expression is: \(-\frac{\sin P}{4}\) 1. Simplified expression: \( -\frac{\sin P}{4} \) The answers provided in the question seem to reflect that they wanted the simplified expression as \( -\frac{\sin P}{4} \) represented among the options. Just as a bonus trivia, did you know that trigonometric identities have been crucial in advancements in mathematics since ancient times? The ancient Greeks utilized them in astronomy to predict astronomical events! And if you find these concepts fascinating, why not delve deeper into trigonometric systems used in various applications? These tools are utilized not just in triangle-related problems, but also in engineering, physics, and even in computer science for graphics and animation!