Concept:
Many legendary Indian musicians, poets, and dance gurus made remarkable contributions to Indian classical music and dance traditions. Their names are closely associated with particular artistic achievements and schools of performance.
Step 1: Match Swami Haridas with his contribution.
\[
\text{Swami Haridas}
\]
was a great saint musician and an important:
\[
\text{Patron of Dhrupad Singing}
\]
Thus:
\[
A \rightarrow IV
\]
Step 2: Match Shri Jayadeva with his contribution.
\[
\text{Shri Jayadeva}
\]
wrote the famous:
\[
\textit{Gita Govinda}
\]
which contains:
\[
\text{Ashtapadis}
\]
Thus:
\[
B \rightarrow III
\]
Step 3: Match Shri Kitappa Pillai with his contribution.
\[
\text{Shri Kitappa Pillai}
\]
was a highly respected:
\[
\text{Guru of Bharata Natyam}
\]
Thus:
\[
C \rightarrow II
\]
Step 4: Match Shri Tyagaraja with his contribution.
\[
\text{Shri Tyagaraja}
\]
is one of the greatest saint composers of:
\[
\text{Carnatic Music}
\]
Thus:
\[
D \rightarrow I
\]
Step 5: Write the final matching sequence.
The correct matching is:
\[
A\text{-}IV,\ B\text{-}III,\ C\text{-}II,\ D\text{-}I
\]
Step 6: Compare with the given options.
Among the given choices:
\[
(A)\ A\text{-}I,\ B\text{-}III,\ C\text{-}II,\ D\text{-}IV
\]
\[
(B)\ A\text{-}III,\ B\text{-}II,\ C\text{-}I,\ D\text{-}IV
\]
\[
(C)\ A\text{-}IV,\ B\text{-}III,\ C\text{-}II,\ D\text{-}I
\]
\[
(D)\ A\text{-}II,\ B\text{-}IV,\ C\text{-}I,\ D\text{-}III
\]
the correct answer is:
\[
\boxed{A\text{-}IV,\ B\text{-}III,\ C\text{-}II,\ D\text{-}I}
\]
Hence:
\[
\boxed{\mathrm{(C)}}
\]