If is true that SQRT(Variance) = SD then one cannot have n-1 in the denominator and the other not but should be same?
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Using $n-1$ as denominator instead of $n$ is called 'Bessel's correction' and it is used to reduce bias in the estimation of variance and standard deviation of the population.
If the sample variance is computed with $n$ as denominator, the sample variance is a biased estimator of the population variance.
As you correctly said, it is coherent to use the same denominator in computing the sample variance and the sample standard error.