what is cotangent of infinity

If you look at this cotangent graph perhaps you will able to 'see' why    $$cot(\infty)$$    is undefined.

https://www.desmos.com/calculator/g0tp6w7r6x

The cotangent of infinity is undefined.

If you mean, "What angle has a cotangent value of (or approaching) infinity?", then it will be 0°  plus or minus all whole number multiples of 180°:   ..., -360°, -180°, 0°, 180°, 360°, ...

According to my ti-89 graphing calculator cosign of infinity is sine of infinity.