To express the functions cos(1/x) and tan(1/y) solely in terms of x and y, we start by breaking each function down to understand their dependencies on the variables.
1. **Understanding the Functions**:
- The cosine function cos(1/x) involves the variable x. Here, 1/x transforms the input, with x influencing the angle in radians that the cosine function will evaluate.
- The tangent function tan(1/y) is similar, with y determining the angle as well. The 1/y term serves as the input for the tangent function.
2. **Writing in Terms of x and y**:
Since the expressions cos(1/x) and tan(1/y) already depend only on x and y via the respective transformations, we can simply state:
- cos(1/x) is a function of x and can be rewritten as:
cos(x'), wherex' = 1/x. - tan(1/y) can also be expressed similarly as:
tan(y'), wherey' = 1/y.
3. **Final Expressions**:
Thus, the expressions can be succinctly represented as:
- For cosine:
cos(1/x) - For tangent:
tan(1/y)
These final forms demonstrate the relationship between the trigonometric functions and the variables x and y, giving us expressions that solely rely on these variables while maintaining the mathematical integrity of the original functions.