Without using a calculator, simplify the expression given above to a single trigonometric term in terms of \( \cos 2 x \).

It turns out that the given expression can be written in the form of a well‐known double‐angle formula. (Although the original problem statement omitted the explicit expression, one common example that simplifies in this way is)   8 cos⁴x – 8 cos²x + 1. To show how one might simplify such an expression in terms of cos 2x without a calculator, follow these steps: Step 1. Recall the double‐angle identity:   cos 2x = 2 cos²x – 1. This can be solved for cos²x:   cos²x = (1 + cos 2x)/2. Step 2. Substitute for cos²x in the expression. In our example, we have cos⁴x = (cos²x)² so that   cos⁴x = [(1 + cos 2x)/2]² = (1 + 2 cos 2x + cos² 2x)/4. Step 3. Replace cos⁴x and cos²x in the expression:   8 cos⁴x – 8 cos²x + 1 = 8·[(1 + 2 cos 2x + cos² 2x)/4] – 8·[(1 + cos 2x)/2] + 1. Simplify term‐by‐term:   = 2 (1 + 2 cos 2x + cos² 2x) – 4 (1 + cos 2x) + 1. Step 4. Distribute and combine like terms:   = (2 + 4 cos 2x + 2 cos² 2x) – (4 + 4 cos 2x) + 1   = 2 cos² 2x + (2 + 4 cos 2x – 4 – 4 cos 2x + 1)   = 2 cos² 2x – 1. Step 5. Recognize that 2 cos²2x – 1 is itself the double‐angle formula for cosine:   cos 4x = 2 cos²2x – 1. Thus, we have expressed the original expression as a single trigonometric term:   8 cos⁴x – 8 cos²x + 1 = 2 cos²2x – 1 = cos 4x. So, if the expression to be simplified is indeed 8 cos⁴x – 8 cos²x + 1, then the final answer written in terms of cos 2x is   2 cos²2x – 1. This example illustrates the process of rewriting a trigonometric expression solely in terms of cos 2x using known identities.