How can we express the product of tan(1/x) and cos(1/y) solely in terms of x and y?

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To express the product tan(1/x) * cos(1/y) in terms of x and y only, we will first recall the definitions of the tangent and cosine functions.

The tangent function is defined as:

  • tan(a) = sin(a) / cos(a)

This means:

  • tan(1/x) = sin(1/x) / cos(1/x)

The cosine function can also be expressed with respect to its angle:

  • cos(b) = adjacent / hypotenuse

While we cannot fully eliminate sin(1/x) and cos(1/y) from this expression without assigning specific numerical values or relationships to x and y, we can represent them as follows:

Thus, combining the two, we can write:

tan(1/x) * cos(1/y) = (sin(1/x) / cos(1/x)) * cos(1/y)

Now, substituting

  • tan(1/x) * cos(1/y) = sin(1/x) * (cos(1/y) / cos(1/x))

In conclusion, the expression tan(1/x) * cos(1/y) can be rewritten in terms of x and y as:

sin(1/x) * (cos(1/y) / cos(1/x))

It is important to note that while we expressed the original functions as relationships of sine and cosine, we still maintain x and y into the equation. The full expression cannot be simplified further without additional numerical relationships.

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