We know that in the interval (0,π2), the function sin x is increasing and the function cos x is decreasing.
Therefore
0<sin α1<sin α2<……<sin αn,
and cos α1>cos α2>……>cos αn>0
Then
n sin α1<sin α2+……+sin αn……(1)
and n cos α1>cos α2+……+cos αn>n cos αn
or 1n cos α1<1cos α1+cos α2+……+cos αn<1n cos αn……(2)
Now multiplying (1) and (2), we get
tan α1<sin α1+sin α2+……+sin αncos α1+cos α2+……+cos αn<tan αn