What is the significance of adding two significant trend? Consider significant time-series (a) and (b) from a p-value < 0.05.
If I add both time series together to obtain time-series (c). Will time-series (c) always be significant or can it be insignificant from the individual variations of (a) and (b)

This question is not actually specific to time series. For convenience I will use the same examples (time series a,b,c) in the question but I will ignore periodicity in the answer; including it makes no difference.
Imagine that time series (a) increases by 1 unit every time step i.e. it is a perfectly straight line with no error.
Now imagine that time series (b) decreases by 1 unit every time step i.e. again, a perfectly straight line with no error.
If time series (c) is simply the sum of (a) and (b), then it is simply a flat line.
(a) and (b) would give you significant regression results. (c) would not.
