График функции
0 -250000 -200000 -150000 -100000 -50000 50000 100000 150000 200000 250000 0.0 1.0
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0sin ( cos 2 ( x ) ) = 0 \sin{\left (\cos^{2}{\left (x \right )} \right )} = 0 sin ( cos 2 ( x ) ) = 0 Решаем это уравнение Аналитическое решение x 1 = π 2 x_{1} = \frac{\pi}{2} x 1 = 2 π x 2 = 3 π 2 x_{2} = \frac{3 \pi}{2} x 2 = 2 3 π Численное решение x 1 = − 1.57079642948 x_{1} = -1.57079642948 x 1 = − 1.57079642948 x 2 = 17.2787597998 x_{2} = 17.2787597998 x 2 = 17.2787597998 x 3 = 4.71238877546 x_{3} = 4.71238877546 x 3 = 4.71238877546 x 4 = 76.9690198091 x_{4} = 76.9690198091 x 4 = 76.9690198091 x 5 = − 92.6769828463 x_{5} = -92.6769828463 x 5 = − 92.6769828463 x 6 = − 48.6946859168 x_{6} = -48.6946859168 x 6 = − 48.6946859168 x 7 = − 58.1194639995 x_{7} = -58.1194639995 x 7 = − 58.1194639995 x 8 = 45.5530936922 x_{8} = 45.5530936922 x 8 = 45.5530936922 x 9 = 70.6858345072 x_{9} = 70.6858345072 x 9 = 70.6858345072 x 10 = 92.6769830846 x_{10} = 92.6769830846 x 10 = 92.6769830846 x 11 = 92.6769833069 x_{11} = 92.6769833069 x 11 = 92.6769833069 x 12 = 281.172542964 x_{12} = 281.172542964 x 12 = 281.172542964 x 13 = 51.8362788997 x_{13} = 51.8362788997 x 13 = 51.8362788997 x 14 = 20.4203521499 x_{14} = 20.4203521499 x 14 = 20.4203521499 x 15 = − 86.3937977708 x_{15} = -86.3937977708 x 15 = − 86.3937977708 x 16 = 381.703506797 x_{16} = 381.703506797 x 16 = 381.703506797 x 17 = − 32.9867231115 x_{17} = -32.9867231115 x 17 = − 32.9867231115 x 18 = 70.6858352006 x_{18} = 70.6858352006 x 18 = 70.6858352006 x 19 = − 54.977871617 x_{19} = -54.977871617 x 19 = − 54.977871617 x 20 = 95.8185760586 x_{20} = 95.8185760586 x 20 = 95.8185760586 x 21 = − 26.7035370064 x_{21} = -26.7035370064 x 21 = − 26.7035370064 x 22 = − 70.6858341659 x_{22} = -70.6858341659 x 22 = − 70.6858341659 x 23 = 61.2610569427 x_{23} = 61.2610569427 x 23 = 61.2610569427 x 24 = 1.57079619636 x_{24} = 1.57079619636 x 24 = 1.57079619636 x 25 = − 26.7035373443 x_{25} = -26.7035373443 x 25 = − 26.7035373443 x 26 = − 95.8185758681 x_{26} = -95.8185758681 x 26 = − 95.8185758681 x 27 = − 89.5353907464 x_{27} = -89.5353907464 x 27 = − 89.5353907464 x 28 = − 80.1106125797 x_{28} = -80.1106125797 x 28 = − 80.1106125797 x 29 = − 10.9955744765 x_{29} = -10.9955744765 x 29 = − 10.9955744765 x 30 = 39.2699085906 x_{30} = 39.2699085906 x 30 = 39.2699085906 x 31 = − 51.8362786898 x_{31} = -51.8362786898 x 31 = − 51.8362786898 x 32 = 61.2610565035 x_{32} = 61.2610565035 x 32 = 61.2610565035 x 33 = 80.1106131458 x_{33} = 80.1106131458 x 33 = 80.1106131458 x 34 = − 61.2610569554 x_{34} = -61.2610569554 x 34 = − 61.2610569554 x 35 = − 70.6858344895 x_{35} = -70.6858344895 x 35 = − 70.6858344895 x 36 = − 98.9601687556 x_{36} = -98.9601687556 x 36 = − 98.9601687556 x 37 = − 58.1194640025 x_{37} = -58.1194640025 x 37 = − 58.1194640025 x 38 = − 54.9778716473 x_{38} = -54.9778716473 x 38 = − 54.9778716473 x 39 = 36.1283155957 x_{39} = 36.1283155957 x 39 = 36.1283155957 x 40 = − 64.4026491934 x_{40} = -64.4026491934 x 40 = − 64.4026491934 x 41 = − 10.9955745786 x_{41} = -10.9955745786 x 41 = − 10.9955745786 x 42 = − 20.4203520389 x_{42} = -20.4203520389 x 42 = − 20.4203520389 x 43 = 54.9778712372 x_{43} = 54.9778712372 x 43 = 54.9778712372 x 44 = 86.3937978884 x_{44} = 86.3937978884 x 44 = 86.3937978884 x 45 = − 7.85398149873 x_{45} = -7.85398149873 x 45 = − 7.85398149873 x 46 = − 45.553093588 x_{46} = -45.553093588 x 46 = − 45.553093588 x 47 = − 4.71238877209 x_{47} = -4.71238877209 x 47 = − 4.71238877209 x 48 = − 92.6769830625 x_{48} = -92.6769830625 x 48 = − 92.6769830625 x 49 = − 39.2699083785 x_{49} = -39.2699083785 x 49 = − 39.2699083785 x 50 = − 29.8451300964 x_{50} = -29.8451300964 x 50 = − 29.8451300964 x 51 = 64.4026493088 x_{51} = 64.4026493088 x 51 = 64.4026493088 x 52 = 17.2787600637 x_{52} = 17.2787600637 x 52 = 17.2787600637 x 53 = − 42.4115006161 x_{53} = -42.4115006161 x 53 = − 42.4115006161 x 54 = − 17.2787598016 x_{54} = -17.2787598016 x 54 = − 17.2787598016 x 55 = 67.544242269 x_{55} = 67.544242269 x 55 = 67.544242269 x 56 = 10.9955739516 x_{56} = 10.9955739516 x 56 = 10.9955739516 x 57 = 58.1194644573 x_{57} = 58.1194644573 x 57 = 58.1194644573 x 58 = − 98.9601687274 x_{58} = -98.9601687274 x 58 = − 98.9601687274 x 59 = 7.85398174035 x_{59} = 7.85398174035 x 59 = 7.85398174035 x 60 = 89.5353908457 x_{60} = 89.5353908457 x 60 = 89.5353908457 x 61 = 73.8274274792 x_{61} = 73.8274274792 x 61 = 73.8274274792 x 62 = 26.7035373526 x_{62} = 26.7035373526 x 62 = 26.7035373526 x 63 = − 32.986723047 x_{63} = -32.986723047 x 63 = − 32.986723047 x 64 = 54.9778710064 x_{64} = 54.9778710064 x 64 = 54.9778710064 x 65 = 23.5619451152 x_{65} = 23.5619451152 x 65 = 23.5619451152 x 66 = − 20.4203522235 x_{66} = -20.4203522235 x 66 = − 20.4203522235 x 67 = − 76.969020186 x_{67} = -76.969020186 x 67 = − 76.969020186 x 68 = 48.6946859298 x_{68} = 48.6946859298 x 68 = 48.6946859298 x 69 = − 14.1371668395 x_{69} = -14.1371668395 x 69 = − 14.1371668395 x 70 = − 23.5619450088 x_{70} = -23.5619450088 x 70 = − 23.5619450088 x 71 = 23.5619444255 x_{71} = 23.5619444255 x 71 = 23.5619444255 x 72 = 83.2522055137 x_{72} = 83.2522055137 x 72 = 83.2522055137 x 73 = − 76.9690201865 x_{73} = -76.9690201865 x 73 = − 76.9690201865 x 74 = 83.2522056453 x_{74} = 83.2522056453 x 74 = 83.2522056453 x 75 = − 83.2522055321 x_{75} = -83.2522055321 x 75 = − 83.2522055321 x 76 = 1.57079653821 x_{76} = 1.57079653821 x 76 = 1.57079653821 x 77 = 42.4115007293 x_{77} = 42.4115007293 x 77 = 42.4115007293 x 78 = 10.9955740943 x_{78} = 10.9955740943 x 78 = 10.9955740943 x 79 = 98.960168066 x_{79} = 98.960168066 x 79 = 98.960168066 x 80 = 98.9601683814 x_{80} = 98.9601683814 x 80 = 98.9601683814 x 81 = − 4.71238846662 x_{81} = -4.71238846662 x 81 = − 4.71238846662 x 82 = 32.9867226656 x_{82} = 32.9867226656 x 82 = 32.9867226656 x 83 = 29.8451303201 x_{83} = 29.8451303201 x 83 = 29.8451303201 x 84 = − 73.82742728 x_{84} = -73.82742728 x 84 = − 73.82742728 x 85 = 32.9867224796 x_{85} = 32.9867224796 x 85 = 32.9867224796 x 86 = − 73.8274267521 x_{86} = -73.8274267521 x 86 = − 73.8274267521 x 87 = 61.2610571172 x_{87} = 61.2610571172 x 87 = 61.2610571172 x 88 = 76.9690195341 x_{88} = 76.9690195341 x 88 = 76.9690195341 x 89 = − 36.1283154194 x_{89} = -36.1283154194 x 89 = − 36.1283154194 x 90 = − 48.6946855663 x_{90} = -48.6946855663 x 90 = − 48.6946855663 x 91 = 39.2699083714 x_{91} = 39.2699083714 x 91 = 39.2699083714 x 92 = 14.1371671038 x_{92} = 14.1371671038 x 92 = 14.1371671038 x 93 = − 67.5442421672 x_{93} = -67.5442421672 x 93 = − 67.5442421672
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:sin ( cos 2 ( 0 ) ) \sin{\left (\cos^{2}{\left (0 \right )} \right )} sin ( cos 2 ( 0 ) ) f ( 0 ) = sin ( 1 ) f{\left (0 \right )} = \sin{\left (1 \right )} f ( 0 ) = sin ( 1 ) (0, sin(1))
Экстремумы функции
Для того, чтобы найти экстремумы, нужно решить уравнениеd d x f ( x ) = 0 \frac{d}{d x} f{\left (x \right )} = 0 d x d f ( x ) = 0 d d x f ( x ) = \frac{d}{d x} f{\left (x \right )} = d x d f ( x ) = Первая производная − 2 sin ( x ) cos ( x ) cos ( cos 2 ( x ) ) = 0 - 2 \sin{\left (x \right )} \cos{\left (x \right )} \cos{\left (\cos^{2}{\left (x \right )} \right )} = 0 − 2 sin ( x ) cos ( x ) cos ( cos 2 ( x ) ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 2 x_{2} = \frac{\pi}{2} x 2 = 2 π x 3 = π x_{3} = \pi x 3 = π x 4 = 3 π 2 x_{4} = \frac{3 \pi}{2} x 4 = 2 3 π (0, sin(1)) pi
(--, 0)
2 (pi, sin(1)) 3*pi
(----, 0)
2 Интервалы возрастания и убывания функции: x 4 = π 2 x_{4} = \frac{\pi}{2} x 4 = 2 π x 4 = 3 π 2 x_{4} = \frac{3 \pi}{2} x 4 = 2 3 π x 4 = 0 x_{4} = 0 x 4 = 0 x 4 = π x_{4} = \pi x 4 = π [3*pi/2, oo) (-oo, pi/2] U [pi, 3*pi/2]
Точки перегибов
Найдем точки перегибов, для этого надо решить уравнениеd 2 d x 2 f ( x ) = 0 \frac{d^{2}}{d x^{2}} f{\left (x \right )} = 0 d x 2 d 2 f ( x ) = 0 d 2 d x 2 f ( x ) = \frac{d^{2}}{d x^{2}} f{\left (x \right )} = d x 2 d 2 f ( x ) = Вторая производная 2 ( − 2 sin 2 ( x ) sin ( cos 2 ( x ) ) cos 2 ( x ) + sin 2 ( x ) cos ( cos 2 ( x ) ) − cos 2 ( x ) cos ( cos 2 ( x ) ) ) = 0 2 \left(- 2 \sin^{2}{\left (x \right )} \sin{\left (\cos^{2}{\left (x \right )} \right )} \cos^{2}{\left (x \right )} + \sin^{2}{\left (x \right )} \cos{\left (\cos^{2}{\left (x \right )} \right )} - \cos^{2}{\left (x \right )} \cos{\left (\cos^{2}{\left (x \right )} \right )}\right) = 0 2 ( − 2 sin 2 ( x ) sin ( cos 2 ( x ) ) cos 2 ( x ) + sin 2 ( x ) cos ( cos 2 ( x ) ) − cos 2 ( x ) cos ( cos 2 ( x ) ) ) = 0 Решаем это уравнение x 1 = − 39.9535297523 x_{1} = -39.9535297523 x 1 = − 39.9535297523 x 2 = − 58.8030856739 x_{2} = -58.8030856739 x 2 = − 58.8030856739 x 3 = − 77.6526415954 x_{3} = -77.6526415954 x 3 = − 77.6526415954 x 4 = − 96.502197517 x_{4} = -96.502197517 x 4 = − 96.502197517 x 5 = 10.3119527051 x_{5} = 10.3119527051 x 5 = 10.3119527051 x 6 = 63.7190278161 x_{6} = 63.7190278161 x 6 = 63.7190278161 x 7 = − 46.2367150595 x_{7} = -46.2367150595 x 7 = − 46.2367150595 x 8 = − 48.0110645482 x_{8} = -48.0110645482 x 8 = − 48.0110645482 x 9 = − 0.887174744321 x_{9} = -0.887174744321 x 9 = − 0.887174744321 x 10 = − 19.7367306659 x_{10} = -19.7367306659 x 10 = − 19.7367306659 x 11 = − 70.0022131233 x_{11} = -70.0022131233 x 11 = − 70.0022131233 x 12 = 43.0951224059 x_{12} = 43.0951224059 x 12 = 43.0951224059 x 13 = 85.7101763912 x_{13} = 85.7101763912 x 13 = 85.7101763912 x 14 = − 13.4535453587 x_{14} = -13.4535453587 x 14 = − 13.4535453587 x 15 = 91.9933616984 x_{15} = 91.9933616984 x 15 = 91.9933616984 x 16 = − 79.4269910841 x_{16} = -79.4269910841 x 16 = − 79.4269910841 x 17 = − 61.9446783275 x_{17} = -61.9446783275 x 17 = − 61.9446783275 x 18 = 4.02876739791 x_{18} = 4.02876739791 x 18 = 4.02876739791 x 19 = − 33.6703444452 x_{19} = -33.6703444452 x 19 = − 33.6703444452 x 20 = − 24.2455664844 x_{20} = -24.2455664844 x 20 = − 24.2455664844 x 21 = 73.1438057769 x_{21} = 73.1438057769 x 21 = 73.1438057769 x 22 = − 98.2765470056 x_{22} = -98.2765470056 x 22 = − 98.2765470056 x 23 = − 17.9623811772 x_{23} = -17.9623811772 x 23 = − 17.9623811772 x 24 = 83.9358269026 x_{24} = 83.9358269026 x 24 = 83.9358269026 x 25 = − 85.7101763912 x_{25} = -85.7101763912 x 25 = − 85.7101763912 x 26 = − 91.9933616984 x_{26} = -91.9933616984 x 26 = − 91.9933616984 x 27 = 19.7367306659 x_{27} = 19.7367306659 x 27 = 19.7367306659 x 28 = 48.0110645482 x_{28} = 48.0110645482 x 28 = 48.0110645482 x 29 = − 80.794234249 x_{29} = -80.794234249 x 29 = − 80.794234249 x 30 = 54.2942498553 x_{30} = 54.2942498553 x 30 = 54.2942498553 x 31 = 65.0862709811 x_{31} = 65.0862709811 x 31 = 65.0862709811 x 32 = 30.5287517916 x_{32} = 30.5287517916 x 32 = 30.5287517916 x 33 = − 22.8783233194 x_{33} = -22.8783233194 x 33 = − 22.8783233194 x 34 = 32.3031012802 x_{34} = 32.3031012802 x 34 = 32.3031012802 x 35 = − 83.9358269026 x_{35} = -83.9358269026 x 35 = − 83.9358269026 x 36 = − 11.67919587 x_{36} = -11.67919587 x 36 = − 11.67919587 x 37 = 96.502197517 x_{37} = 96.502197517 x 37 = 96.502197517 x 38 = 29.1615086266 x_{38} = 29.1615086266 x 38 = 29.1615086266 x 39 = − 26.019915973 x_{39} = -26.019915973 x 39 = − 26.019915973 x 40 = 17.9623811772 x_{40} = 17.9623811772 x 40 = 17.9623811772 x 41 = 41.727879241 x_{41} = 41.727879241 x 41 = 41.727879241 x 42 = 24.2455664844 x_{42} = 24.2455664844 x 42 = 24.2455664844 x 43 = − 35.4446939338 x_{43} = -35.4446939338 x 43 = − 35.4446939338 x 44 = − 57.4358425089 x_{44} = -57.4358425089 x 44 = − 57.4358425089 x 45 = 2.25441790927 x_{45} = 2.25441790927 x 45 = 2.25441790927 x 46 = − 2.25441790927 x_{46} = -2.25441790927 x 46 = − 2.25441790927 x 47 = 90.2190122098 x_{47} = 90.2190122098 x 47 = 90.2190122098 x 48 = 61.9446783275 x_{48} = 61.9446783275 x 48 = 61.9446783275 x 49 = − 90.2190122098 x_{49} = -90.2190122098 x 49 = − 90.2190122098 x 50 = − 99.6437901706 x_{50} = -99.6437901706 x 50 = − 99.6437901706 x 51 = 8.53760321645 x_{51} = 8.53760321645 x 51 = 8.53760321645 x 52 = 87.0774195562 x_{52} = 87.0774195562 x 52 = 87.0774195562 x 53 = 98.2765470056 x_{53} = 98.2765470056 x 53 = 98.2765470056 x 54 = 74.5110489418 x_{54} = 74.5110489418 x 54 = 74.5110489418 x 55 = − 14.8207885236 x_{55} = -14.8207885236 x 55 = − 14.8207885236 x 56 = − 63.7190278161 x_{56} = -63.7190278161 x 56 = − 63.7190278161 x 57 = − 4.02876739791 x_{57} = -4.02876739791 x 57 = − 4.02876739791 x 58 = 51.1526572018 x_{58} = 51.1526572018 x 58 = 51.1526572018 x 59 = − 68.2278636347 x_{59} = -68.2278636347 x 59 = − 68.2278636347 x 60 = 70.0022131233 x_{60} = 70.0022131233 x 60 = 70.0022131233 x 61 = 21.1039738308 x_{61} = 21.1039738308 x 61 = 21.1039738308 x 62 = 76.2853984305 x_{62} = 76.2853984305 x 62 = 76.2853984305 x 63 = − 41.727879241 x_{63} = -41.727879241 x 63 = − 41.727879241 x 64 = 26.019915973 x_{64} = 26.019915973 x 64 = 26.019915973 x 65 = 46.2367150595 x_{65} = 46.2367150595 x 65 = 46.2367150595 x 66 = 52.5199003667 x_{66} = 52.5199003667 x 66 = 52.5199003667 x 67 = 7.1703600515 x_{67} = 7.1703600515 x 67 = 7.1703600515 x 68 = 39.9535297523 x_{68} = 39.9535297523 x 68 = 39.9535297523 x 69 = 68.2278636347 x_{69} = 68.2278636347 x 69 = 68.2278636347 x 70 = 95.134954352 x_{70} = 95.134954352 x 70 = 95.134954352 x 71 = − 55.6614930203 x_{71} = -55.6614930203 x 71 = − 55.6614930203 x 72 = − 36.8119370988 x_{72} = -36.8119370988 x 72 = − 36.8119370988 x 73 = 60.5774351625 x_{73} = 60.5774351625 x 73 = 60.5774351625 Интервалы выпуклости и вогнутости: [98.2765470056, oo) (-oo, -99.6437901706]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ sin ( cos 2 ( x ) ) = sin ( cos 2 ( ∞ ~ ) ) \lim_{x \to -\infty} \sin{\left (\cos^{2}{\left (x \right )} \right )} = \sin{\left (\cos^{2}{\left (\tilde{\infty} \right )} \right )} x → − ∞ lim sin ( cos 2 ( x ) ) = sin ( cos 2 ( ∞ ~ ) ) Возьмём предел y = sin ( cos 2 ( ∞ ~ ) ) y = \sin{\left (\cos^{2}{\left (\tilde{\infty} \right )} \right )} y = sin ( cos 2 ( ∞ ~ ) ) lim x → ∞ sin ( cos 2 ( x ) ) = sin ( cos 2 ( ∞ ~ ) ) \lim_{x \to \infty} \sin{\left (\cos^{2}{\left (x \right )} \right )} = \sin{\left (\cos^{2}{\left (\tilde{\infty} \right )} \right )} x → ∞ lim sin ( cos 2 ( x ) ) = sin ( cos 2 ( ∞ ~ ) ) Возьмём предел y = sin ( cos 2 ( ∞ ~ ) ) y = \sin{\left (\cos^{2}{\left (\tilde{\infty} \right )} \right )} y = sin ( cos 2 ( ∞ ~ ) )
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции sin(cos(x)^2), делённой на x при x->+oo и x ->-oolim x → − ∞ ( 1 x sin ( cos 2 ( x ) ) ) = ∞ ~ sin ( ∞ ~ ) cos ( ∞ ~ ) cos ( cos 2 ( ∞ ~ ) ) \lim_{x \to -\infty}\left(\frac{1}{x} \sin{\left (\cos^{2}{\left (x \right )} \right )}\right) = \tilde{\infty} \sin{\left (\tilde{\infty} \right )} \cos{\left (\tilde{\infty} \right )} \cos{\left (\cos^{2}{\left (\tilde{\infty} \right )} \right )} x → − ∞ lim ( x 1 sin ( cos 2 ( x ) ) ) = ∞ ~ sin ( ∞ ~ ) cos ( ∞ ~ ) cos ( cos 2 ( ∞ ~ ) ) Возьмём предел y = ∞ ~ x sin ( ∞ ~ ) cos ( ∞ ~ ) cos ( cos 2 ( ∞ ~ ) ) y = \tilde{\infty} x \sin{\left (\tilde{\infty} \right )} \cos{\left (\tilde{\infty} \right )} \cos{\left (\cos^{2}{\left (\tilde{\infty} \right )} \right )} y = ∞ ~ x sin ( ∞ ~ ) cos ( ∞ ~ ) cos ( cos 2 ( ∞ ~ ) ) lim x → ∞ ( 1 x sin ( cos 2 ( x ) ) ) = ∞ ~ sin ( ∞ ~ ) cos ( ∞ ~ ) cos ( cos 2 ( ∞ ~ ) ) \lim_{x \to \infty}\left(\frac{1}{x} \sin{\left (\cos^{2}{\left (x \right )} \right )}\right) = \tilde{\infty} \sin{\left (\tilde{\infty} \right )} \cos{\left (\tilde{\infty} \right )} \cos{\left (\cos^{2}{\left (\tilde{\infty} \right )} \right )} x → ∞ lim ( x 1 sin ( cos 2 ( x ) ) ) = ∞ ~ sin ( ∞ ~ ) cos ( ∞ ~ ) cos ( cos 2 ( ∞ ~ ) ) Возьмём предел y = ∞ ~ x sin ( ∞ ~ ) cos ( ∞ ~ ) cos ( cos 2 ( ∞ ~ ) ) y = \tilde{\infty} x \sin{\left (\tilde{\infty} \right )} \cos{\left (\tilde{\infty} \right )} \cos{\left (\cos^{2}{\left (\tilde{\infty} \right )} \right )} y = ∞ ~ x sin ( ∞ ~ ) cos ( ∞ ~ ) cos ( cos 2 ( ∞ ~ ) )
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).sin ( cos 2 ( x ) ) = sin ( cos 2 ( x ) ) \sin{\left (\cos^{2}{\left (x \right )} \right )} = \sin{\left (\cos^{2}{\left (x \right )} \right )} sin ( cos 2 ( x ) ) = sin ( cos 2 ( x ) ) sin ( cos 2 ( x ) ) = − sin ( cos 2 ( x ) ) \sin{\left (\cos^{2}{\left (x \right )} \right )} = - \sin{\left (\cos^{2}{\left (x \right )} \right )} sin ( cos 2 ( x ) ) = − sin ( cos 2 ( x ) ) 