To solve the integral \(\int^{\pi}_{0}\frac{\cos^{2}\theta-1}{\sin^{2}\theta}d\theta\), we will break it down step by step. ### Step 1: Simplifying the Integrand The integrand is \(\frac{\cos^{2}\theta - 1}{\sin^{2}\theta}\). We can use the Pythagorean identity, which states that \(\cos^{2}\theta + \sin^{2}\theta = 1\). From this, we can express \(\cos^{2}\theta\) as: \[ \cos^{2}\theta = 1 - \sin^{2}\theta \] Substituting this into our integrand gives: \[ \cos^{2}\theta - 1 = (1 - \sin^{2}\theta) - 1 = -\sin^{2}\theta \] Thus, the integrand simplifies to: \[ \frac{\cos^{2}\theta - 1}{\sin^{2}\theta} = \frac{-\sin^{2}\theta}{\sin^{2}\theta} = -1 \] ### Step 2: Evaluating the Integral Now, we can rewrite the integral: \[ \int^{\pi}_{0} \frac{\cos^{2}\theta - 1}{\sin^{2}\theta} d\theta = \int^{\pi}_{0} -1 \, d\theta \] This integral is straightforward to evaluate: \[ \int^{\pi}_{0} -1 \, d\theta = -\int^{\pi}_{0} 1 \, d\theta = -\left[\theta\right]^{\pi}_{0} = -(\pi - 0) = -\pi \] ### Step 3: Conclusion Thus, the value of the integral is: \[ \int^{\pi}_{0}\frac{\cos^{2}\theta-1}{\sin^{2}\theta}d\theta = -\pi \] ### Step 4: Analyzing the Options Now, let's look at the options provided: - **A. \(\pi\)**: This is incorrect because the integral evaluates to \(-\pi\), not \(\pi\). - **B. \(\frac{\pi}{2}\)**: This is also incorrect for the same reason; the integral does not yield a positive value. - **C. \(-\frac{\pi}{2}\)**: This is incorrect as well; the integral evaluates to \(-\pi\), not \(-\frac{\pi}{2}\). - **D. \(-\pi\)**: This is the correct answer, as we have shown through our calculations. ### Summary of Key Points - The integral simplifies to \(-1\) over the interval from \(0\) to \(\pi\). - The evaluation of the integral yields \(-\pi\). - The correct answer is option D: \(-\pi\). - Understanding the Pythagorean identity is crucial for simplifying trigonometric integrals. This thorough breakdown should help you understand how to approach similar problems in the future!