1. \( \frac{\cos 3 x}{\cos x}=2 \cos 2 x-1 \)

We begin with the given equation   (cos 3x) / (cos x) = 2 cos 2x − 1         (1) Step 1. Express cos 3x in terms of cos x. Recall the triple-angle formula for cosine:   cos 3x = 4 cos³x − 3 cos x. Substitute this into the left side of (1):   (cos 3x) / (cos x) = (4 cos³x − 3 cos x) / (cos x). Assuming cos x ≠ 0 (to avoid division by zero), we can cancel cos x:   4 cos²x − 3. So the equation (1) becomes   4 cos²x − 3 = 2 cos 2x − 1.         (2) Step 2. Express cos 2x in terms of cos x. Recall the double-angle formula:   cos 2x = 2 cos²x − 1. Now substitute it into the right side of (2):   2 cos 2x − 1 = 2(2 cos²x − 1) − 1 = 4 cos²x − 2 − 1 = 4 cos²x − 3. Thus equation (2) becomes   4 cos²x − 3 = 4 cos²x − 3, which is true for all x. Step 3. Conclusion on the Domain Since we divided by cos x, we must remember that cos x ≠ 0. Therefore, the equation   (cos 3x)/(cos x) = 2 cos 2x − 1 is an identity for all real x such that cos x ≠ 0. Answer Summary: The equation is an identity (always true) for all x with cos x ≠ 0.