How do i prove sin z is continuous in the complex region?
If you mean using the definition of continuity, the same way you would in any region- show that, for any complex number, , given any , there exist a real number so that if , then .
Exactly how you would do that, depends on exactly how you are defining sin(z) for z complex. If you are defining sin(z) in terms of its Taylor series, just subtract the Taylor series for sin(z) and . If you defining sin(z) as , are you allowed to use the fact that is continuous?
Or, since sin(z) satisfies the Cauchy-Riemann equations, it follows that it is differentiable and therefore continuous.