We consider the problems of maintaining an approximate maximum matching and an approximate minimum vertex cover in a dynamic graph undergoing a sequence of edge insertions/deletions. Starting with the seminal work of Onak and Rubinfeld [STOC 2010], this problem has received significant attention in recent years. Very recently, extending the framework of Baswana, Gupta and Sen [FOCS 2011], Solomon [FOCS 2016] gave a randomized dynamic algorithm for this problem that has an approximation ratio of $2$ and an amortised update time of $O(1)$ with high probability. This algorithm requires the assumption of an {\em oblivious adversary}, meaning that the future sequence of edge insertions/deletions in the graph cannot depend in any way on the algorithm's past output. A natural way to remove the assumption on oblivious adversary is to give a deterministic dynamic algorithm for the same problem in $O(1)$ update time. In this paper, we resolve this question. We present a new {\em deterministic} fully dynamic algorithm that maintains a $O(1)$-approximate minimum vertex cover and maximum fractional matching, with an amortised update time of $O(1)$. Previously, the best deterministic algorithm for this problem was due to Bhattacharya, Henzinger and Italiano [SODA 2015]; it had an approximation ratio of $(2+ε)$ and an amortised update time of $O(\log n/ε^2)$. Our results also extend to a fully dynamic $O(f^3)$-approximate algorithm with $O(f^2)$ amortized update time for the hypergraph vertex cover and fractional hypergraph matching problems, where every hyperedge has at most $f$ vertices.