Solving a Trigonometric Equation In Exercises 25-38, find all solutions of the equation in the interval \( \mathbf{0 , 2 \pi )} \). \( \begin{array}{ll}\text { 25. } \cos ^{3} x=\cos x & \text { 26. } \sec ^{2} x-1=0\end{array} \)

Sure, let's solve each trigonometric equation step-by-step within the interval \( [0, 2\pi) \). --- ### **Problem 25:** \[ \cos^3 x = \cos x \] **Step 1: Bring all terms to one side of the equation.** \[ \cos^3 x - \cos x = 0 \] **Step 2: Factor out the common term \(\cos x\).** \[ \cos x (\cos^2 x - 1) = 0 \] **Step 3: Use the Zero Product Property.** \[ \cos x = 0 \quad \text{or} \quad \cos^2 x - 1 = 0 \] **Step 4: Solve each equation separately.** 1. **When \(\cos x = 0\):** \[ x = \frac{\pi}{2}, \quad \frac{3\pi}{2} \] 2. **When \(\cos^2 x - 1 = 0\):** \[ \cos^2 x = 1 \implies \cos x = \pm 1 \] - **\(\cos x = 1\):** \[ x = 0 \] - **\(\cos x = -1\):** \[ x = \pi \] **Step 5: Combine all solutions within \( [0, 2\pi) \).** \[ x = 0, \quad \frac{\pi}{2}, \quad \pi, \quad \frac{3\pi}{2} \] --- ### **Problem 26:** \[ \sec^2 x - 1 = 0 \] **Step 1: Recall the identity involving secant and cosine.** \[ \sec x = \frac{1}{\cos x} \implies \sec^2 x = \frac{1}{\cos^2 x} \] **Step 2: Substitute and solve the equation.** \[ \sec^2 x - 1 = 0 \implies \frac{1}{\cos^2 x} - 1 = 0 \] \[ \frac{1 - \cos^2 x}{\cos^2 x} = 0 \] \[ 1 - \cos^2 x = 0 \implies \cos^2 x = 1 \implies \cos x = \pm 1 \] **Step 3: Find all angles \( x \) in \( [0, 2\pi) \) where \(\cos x = 1\) or \(\cos x = -1\).** 1. **\(\cos x = 1\):** \[ x = 0 \] 2. **\(\cos x = -1\):** \[ x = \pi \] **Step 4: Combine all solutions within \( [0, 2\pi) \).** \[ x = 0, \quad \pi \] --- ### **Summary of Solutions:** - **Problem 25:** \[ x = 0, \quad \frac{\pi}{2}, \quad \pi, \quad \frac{3\pi}{2} \] - **Problem 26:** \[ x = 0, \quad \pi \]