To solve the problem where we need to calculate \( \sin^2 \theta - \cos^2 \theta \) given that \( \tan \theta = \frac{4}{3} \), we will follow a step-by-step approach. ### Step 1: Understanding the relationship between sine, cosine, and tangent The tangent function is defined as the ratio of the sine and cosine functions: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \] Given \( \tan \theta = \frac{4}{3} \), we can express this as: \[ \sin \theta = 4k \quad \text{and} \quad \cos \theta = 3k \] for some positive constant \( k \). ### Step 2: Using the Pythagorean identity We know from the Pythagorean identity that: \[ \sin^2 \theta + \cos^2 \theta = 1 \] Substituting our expressions for \( \sin \theta \) and \( \cos \theta \): \[ (4k)^2 + (3k)^2 = 1 \] This simplifies to: \[ 16k^2 + 9k^2 = 1 \] \[ 25k^2 = 1 \] From this, we can solve for \( k^2 \): \[ k^2 = \frac{1}{25} \] ### Step 3: Finding \( \sin^2 \theta \) and \( \cos^2 \theta \) Now we can find \( \sin^2 \theta \) and \( \cos^2 \theta \): \[ \sin^2 \theta = (4k)^2 = 16k^2 = 16 \cdot \frac{1}{25} = \frac{16}{25} \] \[ \cos^2 \theta = (3k)^2 = 9k^2 = 9 \cdot \frac{1}{25} = \frac{9}{25} \] ### Step 4: Calculating \( \sin^2 \theta - \cos^2 \theta \) Now we can calculate \( \sin^2 \theta - \cos^2 \theta \): \[ \sin^2 \theta - \cos^2 \theta = \frac{16}{25} - \frac{9}{25} = \frac{16 - 9}{25} = \frac{7}{25} \] ### Conclusion: Final Answer Thus, the final answer is: \[ \sin^2 \theta - \cos^2 \theta = \frac{7}{25} \] The correct option is **C**. ### Explanation of Other Options - **Option A: \( \frac{16}{25} \)**: This represents \( \sin^2 \theta \) alone, not the difference. - **Option B: \( \frac{24}{25} \)**: This does not correspond to any relevant calculation from \( \sin^2 \theta \) or \( \cos^2 \theta \). - **Option D: \( \frac{9}{25} \)**: This represents \( \cos^2 \theta \) alone, not the difference. ### Revision Summary - **Tangent Definition**: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). - **Pythagorean Identity**: \( \sin^2 \theta + \cos^2 \theta = 1 \). - **Calculating Squares**: Use \( k \) to express sine and cosine in terms of a common variable. - **Final Calculation**: \( \sin^2 \theta - \cos^2 \theta = \frac{7}{25} \).