To evaluate the expression tan(x) sin(x) sin(y) dx cos(x) cos(y) dy and determine when it’s equal to zero, we first need to understand the individual components involved in the expression.
The expression consists of:
- tan(x): The tangent function, which can be expressed as sin(x)/cos(x).
- sin(x) sin(y): The product of sine functions, which vary between -1 and 1.
- cos(x) cos(y): The product of cosine functions, which also vary between -1 and 1.
Now, when evaluating the integral, the product will yield zero if any of the following conditions hold true:
- tan(x) = 0, which occurs when x = nπ, where n is any integer.
- sin(x) = 0, which occurs when x = nπ.
- sin(y) = 0, which occurs when y = nπ.
- cos(x) = 0, which occurs when x = (2n+1)π/2.
- cos(y) = 0, which occurs when y = (2n+1)π/2.
Thus, to find when this overall expression equals zero, we set each part to zero to observe that the integral will be zero whenever:
- x is any integer multiple of π,
- y is any integer multiple of π.
In conclusion, this integral is 0 based on the periodic nature of trigonometric functions and the principles of integration, specifically when any component that makes up the integral is equal to zero.