How can I simplify the expression cot(x) * sin(x) * sin(pi/2 – x) * cos(x)?

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To simplify the expression cot(x) * sin(x) * sin(pi/2 – x) * cos(x), we can follow these steps:

  1. Recall Definitions: Start by recalling that:
    • cot(x) = cos(x) / sin(x)
    • sin(pi/2 – x) = cos(x) (This is a fundamental identity for sine and cosine.)
  2. Substituting the Identities: Replace sin(pi/2 – x) in the expression:
    3. cot(x) * sin(x) * cos(x) * cos(x)
  3. Now, Substitute cot(x): Replace cot(x) as:
    4. (cos(x) / sin(x)) * sin(x) * cos(x) * cos(x)
  4. Simplifying the Expression: Upon substitution, you can simplify further:
    5. cos(x) * cos(x) = cos^2(x)

This reduces the original expression to:

cos^2(x)

So, the simplified form of the expression cot(x) * sin(x) * sin(pi/2 – x) * cos(x) is cos²(x).

Thus, the final answer is:

cos²(x)

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