In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative$$- 2 x \sin{\left(x \right)} \cos{\left(x \right)} + \cos^{2}{\left(x \right)} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 95.8185759344887$$
$$x_{2} = 37.7123693157661$$
$$x_{3} = 50.2754273458806$$
$$x_{4} = -0.653271187094403$$
$$x_{5} = -81.6875298021918$$
$$x_{6} = 94.253084424113$$
$$x_{7} = 12.6060134442754$$
$$x_{8} = 6.36162039206566$$
$$x_{9} = 42.4115008234622$$
$$x_{10} = 4.71238898038469$$
$$x_{11} = -39.2699081698724$$
$$x_{12} = -59.6986356231676$$
$$x_{13} = -7.85398163397448$$
$$x_{14} = 81.6875298021918$$
$$x_{15} = 51.8362787842316$$
$$x_{16} = 67.5442420521806$$
$$x_{17} = 78.5461819355535$$
$$x_{18} = -84.8288957966139$$
$$x_{19} = 45.553093477052$$
$$x_{20} = 23.5619449019235$$
$$x_{21} = -36.1283155162826$$
$$x_{22} = 20.4203522483337$$
$$x_{23} = -23.5619449019235$$
$$x_{24} = 70.6858347057703$$
$$x_{25} = -83.2522053201295$$
$$x_{26} = 36.1283155162826$$
$$x_{27} = -80.1106126665397$$
$$x_{28} = 86.3937979737193$$
$$x_{29} = -22.013857636623$$
$$x_{30} = 56.5575080935408$$
$$x_{31} = -15.7397193560049$$
$$x_{32} = -14.1371669411541$$
$$x_{33} = -42.4115008234622$$
$$x_{34} = -37.7123693157661$$
$$x_{35} = -3.29231002128209$$
$$x_{36} = 28.2920048800691$$
$$x_{37} = 92.6769832808989$$
$$x_{38} = 9.4774857054208$$
$$x_{39} = -64.4026493985908$$
$$x_{40} = -17.2787595947439$$
$$x_{41} = 65.9810235167388$$
$$x_{42} = 43.9936619344429$$
$$x_{43} = 89.5353906273091$$
$$x_{44} = 22.013857636623$$
$$x_{45} = 58.1194640914112$$
$$x_{46} = -58.1194640914112$$
$$x_{47} = -31.43183263459$$
$$x_{48} = -94.253084424113$$
$$x_{49} = 73.8274273593601$$
$$x_{50} = 48.6946861306418$$
$$x_{51} = -65.9810235167388$$
$$x_{52} = -51.8362787842316$$
$$x_{53} = -12.6060134442754$$
$$x_{54} = 87.970277977177$$
$$x_{55} = 26.7035375555132$$
$$x_{56} = 7.85398163397448$$
$$x_{57} = -86.3937979737193$$
$$x_{58} = 59.6986356231676$$
$$x_{59} = -72.26355003974$$
$$x_{60} = 34.5719807601687$$
$$x_{61} = -45.553093477052$$
$$x_{62} = -6.36162039206566$$
$$x_{63} = -75.4048544617952$$
$$x_{64} = -73.8274273593601$$
$$x_{65} = 72.26355003974$$
$$x_{66} = 100.535938219808$$
$$x_{67} = -50.2754273458806$$
$$x_{68} = -67.5442420521806$$
$$x_{69} = -20.4203522483337$$
$$x_{70} = -9.4774857054208$$
$$x_{71} = -53.4164352526291$$
$$x_{72} = 64.4026493985908$$
$$x_{73} = -61.261056745001$$
$$x_{74} = 80.1106126665397$$
$$x_{75} = -87.970277977177$$
$$x_{76} = -95.8185759344887$$
$$x_{77} = 14.1371669411541$$
$$x_{78} = 3.29231002128209$$
$$x_{79} = -28.2920048800691$$
$$x_{80} = -97.3945059759883$$
$$x_{81} = 1.5707963267949$$
$$x_{82} = 15.7397193560049$$
$$x_{83} = -102.101761241668$$
$$x_{84} = -43.9936619344429$$
$$x_{85} = -1.5707963267949$$
$$x_{86} = -29.845130209103$$
$$x_{87} = 29.845130209103$$
$$x_{88} = -89.5353906273091$$
The values of the extrema at the points:
(95.8185759344887, 3.676520165044e-28)
(37.7123693157661, 37.7057413561082)
(50.2754273458806, 50.2704552295047)
(-0.653271187094403, -0.411949279841571)
(-81.6875298021918, -81.6844694741999)
(94.253084424113, 94.2504320656642)
(12.6060134442754, 12.5862127897398)
(6.36162039206566, 6.32256349768101)
(42.4115008234622, 4.98859428281591e-28)
(4.71238898038469, 1.59017658143397e-31)
(-39.2699081698724, -2.36773935254175e-30)
(-59.6986356231676, -59.6944482165077)
(-7.85398163397448, -7.36192861774987e-31)
(81.6875298021918, 81.6844694741999)
(51.8362787842316, 3.09398107171563e-30)
(67.5442420521806, 1.3132184568469e-27)
(78.5461819355535, 78.5429992236281)
(-84.8288957966139, -84.8259487900249)
(45.553093477052, 1.74530768724744e-35)
(23.5619449019235, 1.73402701495236e-29)
(-36.1283155162826, -3.66424875021481e-28)
(20.4203522483337, 1.96251734458305e-29)
(-23.5619449019235, -1.73402701495236e-29)
(70.6858347057703, 6.77618297499812e-29)
(-83.2522053201295, -1.7964453843451e-28)
(36.1283155162826, 3.66424875021481e-28)
(-80.1106126665397, -1.92283264304371e-27)
(86.3937979737193, 3.32328051180095e-28)
(-22.013857636623, -22.002507009172)
(56.5575080935408, 56.5530881593697)
(-15.7397193560049, -15.7238519846239)
(-14.1371669411541, -4.29347676987172e-30)
(-42.4115008234622, -4.98859428281591e-28)
(-37.7123693157661, -37.7057413561082)
(-3.29231002128209, -3.21808738200779)
(28.2920048800691, 28.2831712204135)
(92.6769832808989, 2.69152684487792e-27)
(9.4774857054208, 9.45118061522278)
(-64.4026493985908, -2.61429235475567e-27)
(-17.2787595947439, -2.10139136502906e-29)
(65.9810235167388, 65.9772347661069)
(43.9936619344429, 43.9879800316228)
(89.5353906273091, 2.60267852044683e-27)
(22.013857636623, 22.002507009172)
(58.1194640914112, 1.39112146798308e-29)
(-58.1194640914112, -1.39112146798308e-29)
(-31.43183263459, -31.4238809266115)
(-94.253084424113, -94.2504320656642)
(73.8274273593601, 4.43565443427593e-28)
(48.6946861306418, 5.73178094238607e-28)
(-65.9810235167388, -65.9772347661069)
(-51.8362787842316, -3.09398107171563e-30)
(-12.6060134442754, -12.5862127897398)
(87.970277977177, 87.9674362000474)
(26.7035375555132, 1.44419018202913e-29)
(7.85398163397448, 7.36192861774987e-31)
(-86.3937979737193, -3.32328051180095e-28)
(59.6986356231676, 59.6944482165077)
(-72.26355003974, -72.260090646562)
(34.5719807601687, 34.564750982936)
(-45.553093477052, -1.74530768724744e-35)
(-6.36162039206566, -6.32256349768101)
(-75.4048544617952, -75.4015391711531)
(-73.8274273593601, -4.43565443427593e-28)
(72.26355003974, 72.260090646562)
(100.535938219808, 100.533451608344)
(-50.2754273458806, -50.2704552295047)
(-67.5442420521806, -1.3132184568469e-27)
(-20.4203522483337, -1.96251734458305e-29)
(-9.4774857054208, -9.45118061522278)
(-53.4164352526291, -53.4117554551774)
(64.4026493985908, 2.61429235475567e-27)
(-61.261056745001, -5.29879683037424e-28)
(80.1106126665397, 1.92283264304371e-27)
(-87.970277977177, -87.9674362000474)
(-95.8185759344887, -3.676520165044e-28)
(14.1371669411541, 4.29347676987172e-30)
(3.29231002128209, 3.21808738200779)
(-28.2920048800691, -28.2831712204135)
(-97.3945059759883, -97.3919391637355)
(1.5707963267949, 5.8895428941999e-33)
(15.7397193560049, 15.7238519846239)
(-102.101761241668, -2.45314668072882e-27)
(-43.9936619344429, -43.9879800316228)
(-1.5707963267949, -5.8895428941999e-33)
(-29.845130209103, -1.12127665170554e-29)
(29.845130209103, 1.12127665170554e-29)
(-89.5353906273091, -2.60267852044683e-27)
Intervals of increase and decrease of the function:Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
$$x_{1} = 95.8185759344887$$
$$x_{2} = -0.653271187094403$$
$$x_{3} = -81.6875298021918$$
$$x_{4} = 42.4115008234622$$
$$x_{5} = 4.71238898038469$$
$$x_{6} = -59.6986356231676$$
$$x_{7} = 51.8362787842316$$
$$x_{8} = 67.5442420521806$$
$$x_{9} = -84.8288957966139$$
$$x_{10} = 45.553093477052$$
$$x_{11} = 23.5619449019235$$
$$x_{12} = 20.4203522483337$$
$$x_{13} = 70.6858347057703$$
$$x_{14} = 36.1283155162826$$
$$x_{15} = 86.3937979737193$$
$$x_{16} = -22.013857636623$$
$$x_{17} = -15.7397193560049$$
$$x_{18} = -37.7123693157661$$
$$x_{19} = -3.29231002128209$$
$$x_{20} = 92.6769832808989$$
$$x_{21} = 89.5353906273091$$
$$x_{22} = 58.1194640914112$$
$$x_{23} = -31.43183263459$$
$$x_{24} = -94.253084424113$$
$$x_{25} = 73.8274273593601$$
$$x_{26} = 48.6946861306418$$
$$x_{27} = -65.9810235167388$$
$$x_{28} = -12.6060134442754$$
$$x_{29} = 26.7035375555132$$
$$x_{30} = 7.85398163397448$$
$$x_{31} = -72.26355003974$$
$$x_{32} = -6.36162039206566$$
$$x_{33} = -75.4048544617952$$
$$x_{34} = -50.2754273458806$$
$$x_{35} = -9.4774857054208$$
$$x_{36} = -53.4164352526291$$
$$x_{37} = 64.4026493985908$$
$$x_{38} = 80.1106126665397$$
$$x_{39} = -87.970277977177$$
$$x_{40} = 14.1371669411541$$
$$x_{41} = -28.2920048800691$$
$$x_{42} = -97.3945059759883$$
$$x_{43} = 1.5707963267949$$
$$x_{44} = -43.9936619344429$$
$$x_{45} = 29.845130209103$$
Maxima of the function at points:
$$x_{45} = 37.7123693157661$$
$$x_{45} = 50.2754273458806$$
$$x_{45} = 94.253084424113$$
$$x_{45} = 12.6060134442754$$
$$x_{45} = 6.36162039206566$$
$$x_{45} = -39.2699081698724$$
$$x_{45} = -7.85398163397448$$
$$x_{45} = 81.6875298021918$$
$$x_{45} = 78.5461819355535$$
$$x_{45} = -36.1283155162826$$
$$x_{45} = -23.5619449019235$$
$$x_{45} = -83.2522053201295$$
$$x_{45} = -80.1106126665397$$
$$x_{45} = 56.5575080935408$$
$$x_{45} = -14.1371669411541$$
$$x_{45} = -42.4115008234622$$
$$x_{45} = 28.2920048800691$$
$$x_{45} = 9.4774857054208$$
$$x_{45} = -64.4026493985908$$
$$x_{45} = -17.2787595947439$$
$$x_{45} = 65.9810235167388$$
$$x_{45} = 43.9936619344429$$
$$x_{45} = 22.013857636623$$
$$x_{45} = -58.1194640914112$$
$$x_{45} = -51.8362787842316$$
$$x_{45} = 87.970277977177$$
$$x_{45} = -86.3937979737193$$
$$x_{45} = 59.6986356231676$$
$$x_{45} = 34.5719807601687$$
$$x_{45} = -45.553093477052$$
$$x_{45} = -73.8274273593601$$
$$x_{45} = 72.26355003974$$
$$x_{45} = 100.535938219808$$
$$x_{45} = -67.5442420521806$$
$$x_{45} = -20.4203522483337$$
$$x_{45} = -61.261056745001$$
$$x_{45} = -95.8185759344887$$
$$x_{45} = 3.29231002128209$$
$$x_{45} = 15.7397193560049$$
$$x_{45} = -102.101761241668$$
$$x_{45} = -1.5707963267949$$
$$x_{45} = -29.845130209103$$
$$x_{45} = -89.5353906273091$$
Decreasing at intervals
$$\left[95.8185759344887, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -97.3945059759883\right]$$