$$ \cos (2\infty -1)\sin (2\infty ) $$

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The expression $$ \cos(2\infty - 1) \sin(2\infty) $$ is undefined or oscillates between -1 and 1. To evaluate the expression $$ \cos(2\infty - 1) \sin(2\infty) $$, we need to consider the behavior of the trigonometric functions as their arguments approach infinity. 1. **Evaluate $$ \sin(2\infty) $$**: The sine function oscillates between -1 and 1 for all real numbers. As the argument approaches infinity, $$ \sin(2\infty) $$ does not converge to a specific value; it continues to oscillate. Therefore, $$ \sin(2\infty) $$ is undefined in the traditional sense, but we can say it takes values in the interval $$[-1, 1]$$. 2. **Evaluate $$ \cos(2\infty - 1) $$**: Similarly, the cosine function also oscillates between -1 and 1. As $$ 2\infty - 1 $$ approaches infinity, $$ \cos(2\infty - 1) $$ also does not converge to a specific value and continues to oscillate. Thus, $$ \cos(2\infty - 1) $$ is also undefined in the traditional sense, but it takes values in the interval $$[-1, 1]$$. 3. **Combine the results**: The product $$ \cos(2\infty - 1) \sin(2\infty) $$ involves multiplying two oscillating functions. Since both functions can take any value between -1 and 1, the product can also take any value in the range $$[-1, 1]$$. In conclusion, the expression $$ \cos(2\infty - 1) \sin(2\infty) $$ does not have a defined limit or value as it involves oscillating functions. Therefore, we can say: $$ \cos(2\infty - 1) \sin(2\infty) \text{ is undefined or oscillates between } -1 \text{ and } 1. $$

\( \cos (2\infty -1)\sin (2\infty ) \)

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\( \cos (2\infty -1)\sin (2\infty ) \)

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