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$$\frac{- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{x + 1} - \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\left(x + 1\right)^{2}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -80.8928816808707$$
$$x_{2} = -99.7430349489701$$
$$x_{3} = -63.6132585554971$$
$$x_{4} = -47.9039581285518$$
$$x_{5} = -98.172197695036$$
$$x_{6} = 84.0346635398793$$
$$x_{7} = 46.3332101330021$$
$$x_{8} = 38.478177732588$$
$$x_{9} = 19.6228339741551$$
$$x_{10} = 82.4638118824473$$
$$x_{11} = 16.4790625040945$$
$$x_{12} = 41.6202371710741$$
$$x_{13} = -46.33297711484$$
$$x_{14} = 66.7551541995631$$
$$x_{15} = -57.3296278828154$$
$$x_{16} = 62.0424894121024$$
$$x_{17} = 33.7649303669424$$
$$x_{18} = 8.61339783522102$$
$$x_{19} = -5.44173882723211$$
$$x_{20} = 18.0510381254578$$
$$x_{21} = -33.76449140679$$
$$x_{22} = -62.042359483332$$
$$x_{23} = 98.1722495795572$$
$$x_{24} = 32.1937937404494$$
$$x_{25} = 55.7588651168628$$
$$x_{26} = -25.9081038458568$$
$$x_{27} = 24.3374775388136$$
$$x_{28} = -82.4637383451864$$
$$x_{29} = 2.28057021563236$$
$$x_{30} = -60.471454983256$$
$$x_{31} = 85.6055131901373$$
$$x_{32} = 74.6095191089778$$
$$x_{33} = 40.049216384194$$
$$x_{34} = -27.4794955733025$$
$$x_{35} = -11.7577501031099$$
$$x_{36} = 187.70883626286$$
$$x_{37} = 25.9088498373569$$
$$x_{38} = 27.4801585795776$$
$$x_{39} = -13.3315066037056$$
$$x_{40} = 63.6133821448946$$
$$x_{41} = 99.743085211956$$
$$x_{42} = 44.762232510841$$
$$x_{43} = -19.6215319912886$$
$$x_{44} = -3.83985112537054$$
$$x_{45} = 60.4715917520353$$
$$x_{46} = -16.4772142695041$$
$$x_{47} = 11.7613921271159$$
$$x_{48} = -49.4749271716005$$
$$x_{49} = 0.637196330969125$$
$$x_{50} = 30.6226232987428$$
$$x_{51} = -38.4778397936073$$
$$x_{52} = 68.3260341292281$$
$$x_{53} = -79.322022596248$$
$$x_{54} = -90.317989831739$$
$$x_{55} = -41.6199483594113$$
$$x_{56} = -84.0345927265613$$
$$x_{57} = -85.6054449521595$$
$$x_{58} = 88.7472068907202$$
$$x_{59} = 76.1803827297937$$
$$x_{60} = -69.8968079909424$$
$$x_{61} = 22.7660290738215$$
$$x_{62} = 96.6014126819581$$
$$x_{63} = 54.1879434234347$$
$$x_{64} = -93.4596775892801$$
$$x_{65} = -54.1877730844831$$
$$x_{66} = 3.87589679173726$$
$$x_{67} = -68.3259270041136$$
$$x_{68} = 91.8888937560759$$
$$x_{69} = 69.8969103540797$$
$$x_{70} = 10.1878453044909$$
$$x_{71} = -10.1829786980484$$
$$x_{72} = -40.0489044548563$$
$$x_{73} = -77.7511609427056$$
$$x_{74} = -71.4676852032561$$
$$x_{75} = -18.0494987719381$$
$$x_{76} = 52.6170143837707$$
$$x_{77} = 90.3180511337085$$
$$x_{78} = -24.3366319328506$$
$$x_{79} = -76.1802965592094$$
$$x_{80} = 47.9041761074919$$
$$x_{81} = -55.7587042438767$$
$$x_{82} = 77.7512436659628$$
$$x_{83} = -35.3356368025816$$
$$x_{84} = -44.7619828411789$$
$$x_{85} = -91.8888345323567$$
$$x_{86} = -2.15134433588925$$
$$x_{87} = -32.1933108467876$$
The values of the extrema at the points:
(-80.89288168087066, -0.00625825728073336)
(-99.74303494897006, -0.00506358337322892)
(-63.61325855549712, 0.00798527452696611)
(-47.90395812855178, 0.0106594755120283)
(-98.172197695036, 0.00514543658512232)
(84.03466353987932, -0.00587985341438749)
(46.33321013300211, -0.0105628184636869)
(38.47817773258804, 0.0126642092304746)
(19.622833974155125, 0.0242378477533134)
(82.46381188244735, 0.00599051273937728)
(16.479062504094543, 0.0285939565978739)
(41.620237171074066, 0.011730708921835)
(-46.33297711483998, -0.0110288276247335)
(66.75515419956312, 0.00737931146584952)
(-57.32962788281537, 0.00887597384673406)
(62.04248941210236, -0.00793090944828029)
(33.76493036694237, -0.0143808225713452)
(8.61339783522102, -0.0519405415328574)
(-5.441738827232107, -0.111862019279856)
(18.05103812545779, -0.0262362545191566)
(-33.76449140678997, -0.0152586464362218)
(-62.04235948333203, -0.00819075854569188)
(98.17224957955716, 0.00504166888895891)
(32.19379374044943, 0.0150613482038312)
(55.758865116862815, -0.0088088547861832)
(-25.908103845856804, 0.0200697449426835)
(24.337477538813616, -0.0197297727754678)
(-82.46373834518636, 0.00613758455702892)
(2.2805702156323617, -0.150672544934396)
(-60.471454983255995, 0.00840709765728415)
(85.60551319013732, 0.00577320829852196)
(74.60951910897779, -0.00661277936376119)
(40.04921638419398, -0.0121795970061747)
(-27.479495573302465, -0.0188791695413614)
(-11.757750103109899, -0.0464279997867718)
(187.70883626286036, -0.00264957515947118)
(25.908849837356932, 0.0185780406697524)
(27.480158579577623, -0.0175533771136332)
(-13.331506603705582, 0.0405132573447961)
(63.61338214489455, 0.00773810337143086)
(99.74308521195597, -0.0049630586647441)
(44.762232510840974, 0.0109253882656147)
(-19.621531991288602, 0.0268409633779861)
(-3.8398511253705365, 0.173398481994393)
(60.47159175203532, 0.00813356945441207)
(-16.47721426950408, 0.0322887105600465)
(11.761392127115943, -0.0391506392825053)
(-49.4749271716005, -0.0103140620006304)
(0.637196330969125, 0.292082592805874)
(30.622623298742763, -0.015809488870913)
(-38.47783979360729, 0.0133400300463799)
(68.32603412922812, -0.00721211017589145)
(-79.32202259624798, 0.00638377042868822)
(-90.317989831739, -0.00559788869847007)
(-41.619948359411254, 0.0123082905137621)
(-84.03459272656133, -0.00602147754269582)
(-85.6054449521595, 0.00590968192557079)
(88.74720689072021, 0.00557111756370173)
(76.18038272979369, 0.00647819420500065)
(-69.89680799094245, 0.00725703916120515)
(22.766029073821525, 0.0210337781400242)
(96.60141268195807, -0.00512280944940889)
(54.18794342343466, 0.00905957804504919)
(-93.45967758928009, -0.00540768367597939)
(-54.18777308448308, 0.00940024143486964)
(3.8758967917372615, 0.102010295291554)
(-68.3259270041136, -0.00742635466561701)
(91.88889375607587, 0.00538269685829326)
(69.89691035407965, 0.00705231812713097)
(10.187845304490947, 0.0446467933531117)
(-10.182978698048352, 0.0543680322906561)
(-40.048904454856334, -0.0128034069967485)
(-77.75116094270562, -0.00651442120647222)
(-71.46768520325605, -0.0070952722893795)
(-18.049498771938094, -0.0293137725748063)
(52.61701438377071, -0.00932499256816396)
(90.31805113370845, -0.00547528663879163)
(-24.336631932850587, -0.021420626484021)
(-76.18029655920942, 0.00665053166012875)
(47.90417610749193, 0.0102235414010152)
(-55.7587042438767, -0.0091305878135904)
(77.75124366596276, -0.00634897811440877)
(-35.33563680258161, 0.0145605860581051)
(-44.76198284117895, 0.0114246964043575)
(-91.8888345323567, 0.0055011425426788)
(-2.1513443358892483, -0.398334456870323)
(-32.19331084678758, 0.0160270188053562)
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:
$$x_{1} = -80.8928816808707$$
$$x_{2} = -99.7430349489701$$
$$x_{3} = 84.0346635398793$$
$$x_{4} = 46.3332101330021$$
$$x_{5} = -46.33297711484$$
$$x_{6} = 62.0424894121024$$
$$x_{7} = 33.7649303669424$$
$$x_{8} = 8.61339783522102$$
$$x_{9} = -5.44173882723211$$
$$x_{10} = 18.0510381254578$$
$$x_{11} = -33.76449140679$$
$$x_{12} = -62.042359483332$$
$$x_{13} = 55.7588651168628$$
$$x_{14} = 24.3374775388136$$
$$x_{15} = 2.28057021563236$$
$$x_{16} = 74.6095191089778$$
$$x_{17} = 40.049216384194$$
$$x_{18} = -27.4794955733025$$
$$x_{19} = -11.7577501031099$$
$$x_{20} = 187.70883626286$$
$$x_{21} = 27.4801585795776$$
$$x_{22} = 99.743085211956$$
$$x_{23} = 11.7613921271159$$
$$x_{24} = -49.4749271716005$$
$$x_{25} = 30.6226232987428$$
$$x_{26} = 68.3260341292281$$
$$x_{27} = -90.317989831739$$
$$x_{28} = -84.0345927265613$$
$$x_{29} = 96.6014126819581$$
$$x_{30} = -93.4596775892801$$
$$x_{31} = -68.3259270041136$$
$$x_{32} = -40.0489044548563$$
$$x_{33} = -77.7511609427056$$
$$x_{34} = -71.4676852032561$$
$$x_{35} = -18.0494987719381$$
$$x_{36} = 52.6170143837707$$
$$x_{37} = 90.3180511337085$$
$$x_{38} = -24.3366319328506$$
$$x_{39} = -55.7587042438767$$
$$x_{40} = 77.7512436659628$$
$$x_{41} = -2.15134433588925$$
Maxima of the function at points:
$$x_{41} = -63.6132585554971$$
$$x_{41} = -47.9039581285518$$
$$x_{41} = -98.172197695036$$
$$x_{41} = 38.478177732588$$
$$x_{41} = 19.6228339741551$$
$$x_{41} = 82.4638118824473$$
$$x_{41} = 16.4790625040945$$
$$x_{41} = 41.6202371710741$$
$$x_{41} = 66.7551541995631$$
$$x_{41} = -57.3296278828154$$
$$x_{41} = 98.1722495795572$$
$$x_{41} = 32.1937937404494$$
$$x_{41} = -25.9081038458568$$
$$x_{41} = -82.4637383451864$$
$$x_{41} = -60.471454983256$$
$$x_{41} = 85.6055131901373$$
$$x_{41} = 25.9088498373569$$
$$x_{41} = -13.3315066037056$$
$$x_{41} = 63.6133821448946$$
$$x_{41} = 44.762232510841$$
$$x_{41} = -19.6215319912886$$
$$x_{41} = -3.83985112537054$$
$$x_{41} = 60.4715917520353$$
$$x_{41} = -16.4772142695041$$
$$x_{41} = 0.637196330969125$$
$$x_{41} = -38.4778397936073$$
$$x_{41} = -79.322022596248$$
$$x_{41} = -41.6199483594113$$
$$x_{41} = -85.6054449521595$$
$$x_{41} = 88.7472068907202$$
$$x_{41} = 76.1803827297937$$
$$x_{41} = -69.8968079909424$$
$$x_{41} = 22.7660290738215$$
$$x_{41} = 54.1879434234347$$
$$x_{41} = -54.1877730844831$$
$$x_{41} = 3.87589679173726$$
$$x_{41} = 91.8888937560759$$
$$x_{41} = 69.8969103540797$$
$$x_{41} = 10.1878453044909$$
$$x_{41} = -10.1829786980484$$
$$x_{41} = -76.1802965592094$$
$$x_{41} = 47.9041761074919$$
$$x_{41} = -35.3356368025816$$
$$x_{41} = -44.7619828411789$$
$$x_{41} = -91.8888345323567$$
$$x_{41} = -32.1933108467876$$
Decreasing at intervals
$$\left[187.70883626286, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -99.7430349489701\right]$$