Section 7.3 Practice: Prove each Identity. 1) \( \sec (x)-\sin (x) \tan (x)=\cos (x) \)

To prove the identity \( \sec (x) - \sin (x) \tan (x) = \cos (x) \), we can start by rewriting the left side using their definitions. We know that \( \sec(x) = \frac{1}{\cos(x)} \) and \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). Therefore, the expression becomes: \[ \frac{1}{\cos(x)} - \sin(x) \cdot \frac{\sin(x)}{\cos(x)} = \frac{1}{\cos(x)} - \frac{\sin^2(x)}{\cos(x)} \] Combining the two fractions over a common denominator, we get: \[ \frac{1 - \sin^2(x)}{\cos(x)} \] Using the Pythagorean identity \( 1 - \sin^2(x) = \cos^2(x) \), we can substitute to simplify: \[ \frac{\cos^2(x)}{\cos(x)} = \cos(x) \] Thus, we’ve proven \( \sec(x) - \sin(x) \tan(x) = \cos(x) \). Now, let's remember that this identity is equal to a specific form of the relationship between sine, cosine, and secant functions, showcasing how they can transform into each other through simple manipulations. Understanding trigonometric identities can help you identify them in complex expressions, making problem-solving easier! When dealing with trigonometric functions, it’s crucial to keep fundamental identities handy for quick simplifications or proofs. Common mistakes often stem from misapplying the identities or neglecting to recognize equivalent forms. Always double-check your steps for clarity!