1. groupby.head(1)

The relevant groupby method to drop duplicates in each group is groupby.head(1). Note that it is important to pass 1 to select the first row of each date-cid pair.

df1 = df.groupby(['date', 'cid']).head(1)

2. duplicated() is more flexible

Another method is to use duplicated() to create a boolean mask and filter.

df3 = df[~df.duplicated(['date', 'cid'])]

An advantage of this method over drop_duplicates() is that is can be chained with other boolean masks to filter the dataframe more flexibly. For example, to select the unique cids in Nevada for each date, use:

df_nv = df[df['state'].eq('NV') & ~df.duplicated(['date', 'cid'])]

3. groupby.sample(1)

Another method to select a unique row from each group to use groupby.sample(). Unlike the previous methods mentioned, it selects a row from each group randomly (whereas the others only keep the first row from each group).

df4 = df.groupby(['date', 'cid']).sample(n=1)

You can verify that df1, df2 (ayhan's output) and df3 all produce the very same output and df4 produces an output where size and nunique of cid match for each date (as required in the OP). In short, the following returns True.

w, x, y, z = [d.groupby('date')['cid'].agg(['size', 'nunique']) for d in (df1, df2, df3, df4)]
w.equals(x) and w.equals(y) and w.equals(z)   # True

and w, x, y, z all look like the following:

       size  nunique
date        
2005      7        3
2006    237       10
2007   3610      227
2008   1318       52
2009   2664      142
2010    997       57
2011   6390      219
2012   2904       99
2013   7875      238
2014   3979      146

Rather than removing duplicates during the pivot table process, use the df.drop_duplicates() function to selectively drop duplicates.

For example if you are pivoting using these index='c0' and columns='c1' then this simple step yields the correct counts.

In this example the 5th row is a duplicate of the 4th (ignoring the non-pivoted c2 column

import pandas as pd
data = {'c0':[0,1,0,1,1], 'c1':[0,0,1,1,1], 'person':[0,0,1,1,1], 'c_other':[1,2,3,4,5]}
df = pd.DataFrame(data)
df2 = df.drop_duplicates(subset=['c0','c1','person'])
pd.pivot_table(df2, index='c0',columns='c1',values='person', aggfunc='count')

This correctly outputs

c1  0  1
c0      
0   1  1
1   1  1

Assuming that Stroke only contains 1 or 0, you can do:

result_df = df.groupby('Residence_type').sum()

>>> result_df
                Stroke
Residence_type        
Rural                3
Urban                2

>>> result_df.Stroke['Rural'] - result_df.Stroke['Urban']
1