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#.
