The period of both #sin ktheta and tan ktheta# is #(2pi)/k#

Here; the periods of the separate terms are #(14pi)/15 and 5pi#..

The compounded period for the sum #f(theta)# is given by

#(14/15)piL = 5piM#, for the least multiples L and Ml that get common value as an integer multiple of #pi#..

L = 75/2 and M = 7, and the common integer value is #35pi#.

So, the period of #f(theta) = 35 pi#.

Now, see the effect of the period.

#f(theta+35pi)#

#=tan((15/7)(theta+35pi))-cos((2/5)(theta+35pi#))

#=tan(75pi+(15/7)theta)-cos(14pi+(2/5)theta))=tan ((15/7)theta)#

#-cos((2/5)theta))#

#=f(theta)#

Note that #75pi+_# is in the 3rd quadrant and tangent is positive. Similarly, for the cosine, #14pi+# is in the 1st quadrant and cosine is positive.

The value repeats when #theta# is increased by any integer multiple of #35pi#.