Final answer:
The correct answer is option d. 1.0.
Step-by-step explanation:
The correct answer is 1.0, interpreting 'cos'' as cos2(x) and using the standard trigonometric identity. However, there seems to be a typographical error in the question, and assuming the given expression means to calculate the square of cos(arctan4), the answer remains 1.0.
The expression given is cos' (1.5m + arctan4) cos(arctan4). We can interpret 'cos'' as the square of the cosine function, or cos2(x). Now, using trigonometric identities, we know that cos2(x) = (1 + cos(2x))/2. However, the given expression simplifies directly because cos(arctan4) is a constant value. When we calculate cos(arctan4), we can use a right-angled triangle where the opposite side is 4 and the adjacent side is 1 (from the definition of tan as opposite/adjacent), which gives us a hypotenuse of √(12 + 42) = √17. Thus, cos(arctan4) = 1/√17.
However, the question seems to involve a typographical error as 'cos'' is not standard notation. Assuming it means cos2, we would square the value to get the final result. The square of 1/√17 is 1/17, but since we're squaring cos(arctan4) only and not the entire expression including the factor of 1.5m, the answer to the expression would remain unaffected. This is because the additional term 1.5m doesn't influence the value of the trigonometric function, implying there is an error or missing information in the question as written.
Given the information provided and standard trigonometric identities, the answer should be d. 1.0, as squaring a cosine value where the angle is a real number does not change the maximum value of cosine, which is 1.0.