如果把xcos2xcos3x,看成一个t,那1-cost等价于1/2 *(xcosxcos2x)^2,然后把cosx写成1-1/2x^2,cos2x（见补充 关于等价无穷小的问题。 1-cosx~x^2/2怎么推导出来...

\u5f53x\u65e0\u9650\u8d8b\u5411\u4e8e\u96f6\u65f6,1\uff0dcos2x\u7684\u7b49\u4ef7\u65e0\u7a77\u5c0f\u91cf

\u56e0\u4e3a\u4e0d\u77e5\u9053\u4f60\u76842\u662f\u5728\u4e0b\u9762\u8fd8\u662f\u5728\u6307\u6570\u4f4d\u7f6e\u4e0a\uff0c\u6240\u4ee5\uff1a
\u2460\uff1a1-cos2x~2x²
\u56e0\u4e3ax\u21920\u65f6\uff0c1-cosx~0.5x²\uff0c\u540c\u7406\u53ef\u5f97\u4e0a\u5f0f\u3002
\u2461\uff1a\u6b64\u5916\uff0c1-cos²x=(1+cosx)(1-cosx)~2\u00d70.5x²=x²

\u65b9\u6cd5\u4e00\uff0c\u7528\u6d1b\u5fc5\u8fbe\u6cd5\u5219\uff0c\u5206\u5b50\u662f1-cosx,\u5206\u6bcd\u662fx^2/2\u3002\u65b9\u6cd5\u4e8c\uff0c\u7528\u9ea6\u514b\u52b3\u6797\u5c55\u5f00\u5f0f\u3002\u65b9\u6cd5\u4e09\uff0c\u5c06cosx\u7528\u534a\u89d2\u516c\u5f0f\u5c55\u5f00\u6210x/2\u5f62\u5f0f\u3002\u7b80\u5355\u4e0d\uff0c\u5475\u5475\u3002

﹙cosx﹚﹙g﹙x﹚﹚不能折为cos﹙xg﹙x﹚﹚
分子可以化为1－﹙1－x²/2﹚﹙1－4x²/2﹚﹙1－9x²/2﹚ ﹙高次已略去﹚

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濡傛灉鎶妜cos2xcos3x,鐪嬫垚涓涓猼,閭1-cost绛変环浜1/2 *(xcosxcos2x)^2,鐒 ...
绛旓細锕cosx锕氾箼g锕檟锕氾箽涓嶈兘鎶樹负cos锕檟g锕檟锕氾箽鍒嗗瓙鍙互鍖栦负1锛嶏箼1锛峹²/2锕氾箼1锛4x²/2锕氾箼1锛9x²/2锕 锕欓珮娆″凡鐣ュ幓锕

姹俢osxcos2xcos3x瀵箈鐨勪笉瀹氱Н鍒,鍙婅繖绉嶅緢澶氫笁瑙掔浉涔橀櫎鐨勬柟娉曘俖鐧惧害...
绛旓細鍏堢敤绉寲鍜屽樊鍏紡鍖栫畝寰1/4(1+cos6x+cos4x+cos2x)鍐嶅垎閮ㄧН涔嬪緱1/4x+1/24sin6x+1/16sin4x+1/8sin2x+C

宸茬煡sin2xsin3x=cos2xcos3x,鍒檟鐨勪竴涓彲鑳界殑鍊兼槸?
绛旓細鍥炵瓟锛氬師寮忓嵆鍖栦负cos2xcos3x-sin2xsin3x=0 cos(2x+3x)=0 鎵浠5x=k蟺+蟺/2 鐢ㄥ叕寮:cos(a+b)=cosacosb-sinasinb

cos(3x)涔榗os(2x) 鍖栫畝
绛旓細cos2xcos3x =(1/2)(2cos2xcos3x)=(1/2)(2cos2xcos3x+sin2xsin3x-sin2xsin3x)=(1/2)[(cos2xcos3x+sin2xsin3x)+(cos2xcos3x-sin2xsin3x)]=(1/2)[cos(3x+2x)+cos(3x-2x)]=1/2(cos5x+cosx)鐪嬪湪鎵嬫墦浜嗚繖浜涚殑浠戒笂锛岃閲囩撼~

cosxcos2xcos3x鐨勫鏁 鍘熼1-cosxcos2xcos3x/1-cosx 鍦▁瓒嬭繎浜0鐨勬瀬闄...
绛旓細杩欎釜鏄掓暟鐨勫洓鍒欒繍绠楀拰绗﹀悎鍑芥暟鐨勫鏁 (abc)'=a'bc+ab'c+abc'(cosxcos2xcos3x)'=-sinxcos2xcos3x-2cosxsin2xcos3x-3cosxcox2xsin3x 鏈鍚庣畻鍑虹殑缁撴灉鏄14鍚

cosxcos2xcos3xcos4xcos5x,姹傚,鎬庝箞绠,绠椾笉鍑烘潵鎶,x鎹㈡垚鎺/11銆傛眰鍊...
绛旓細cosx*cos2x*cos4x = 2 sinx*cosx*cos2x*cos4x / (2sinx)= sin2x * cos2x *cos4x /(2sinx) = sin8x / (8sinx)cos3x*cos5x =(1/2) ( cos8x +cos2x)鍘熷紡= (1/16) (1/sinx) [ sin8x cos8x + sin8xcos2x ]= (1/32) (1/sinx) [ sin16x + sin10x +sin6x ]...

cos^3x鍒╃敤鍏紡鍙互鍐欐垚鍏充簬3x鐨勪笁瑙掑嚱鏁,鎬庝箞鍖
绛旓細浣欏鸡鍑芥暟鐨勪笁鍊嶈鍏紡鏄細鎺ㄥ杩囩▼濡備笅锛cos3x=cos(x+2x)=cosx*cos2x-sinx*sin2x =cosx*(2cosx^2-1)-sinx*(2sinxcosx)=2cosx^3-cosx-2cosx(1-cosx^2)=4cosx^3-3cosx

濡傛灉sin2xsin3x=cos2xcos3x(x鍦0鍒90搴﹂棿)姹倄鐨勫(杩囩▼,鏈濂藉啓涓婁负浠...
绛旓細鍘熷紡鍗冲寲涓cos2xcos3x-sin2xsin3x=0 cos锛2x+3x锛=0 鎵浠5x=k蟺+蟺/2 鐢ㄥ叕寮:cos锛坅+b锛=cosacosb-sinasinb

濡備綍鐢ㄧН鍖栧拰宸眰cosx涔樹互cos3x鐨勫?
绛旓細cosx涔樹互cos3x =(cosx)^2*cos2x-sinx*sin2x*cosx =(cos2x+1)/2*cos2x-(sin2x)^2/2 =((cos2x)^2+1-(sin2x)^2)/2杩欎竴姝ュ悗闈㈡妸1鍙樹负 (sin2x)^2+(cos2x)^2 =(cos2x)^2 璁板繂鍙ｈ瘈 绉寲鍜屽樊寰楀拰宸紝浣欏鸡鍦ㄥ悗瑕佺浉鍔狅紱寮傚悕鍑芥暟鍙栨寮︼紝姝ｅ鸡鐩镐箻鍙栬礋鍙枫傝В閲婏細锛1锛夌Н鍖栧拰宸渶鍚...

sin2xsin3x=cos2xcos3x,鍒檟鐨勫煎彲鑳戒负
绛旓細鍘熷紡鍙互鍖栦负cos5x=0 鑰屼綑寮︾殑闆剁偣鏄槸鍦▁=蟺/2 +k蟺 鈭5x=k蟺+蟺/2锛寈=k蟺/5 +蟺/10 锛宬鍙互涓轰换鎰忔暣鏁帮紝鈭翠綘闅忎究鍙栧奸兘琛岋紝鍙婊¤冻杩欎釜寮忓瓙灏卞彲浠

扩展阅读：∫x∧3e∧x∧2dx ... cosx-cos3x sin3x ... cosxcos2xcos3x cosnx ... lim1-cosxcos2xcos3x ... sin5xsin7x ... limx 01cosx cos2x ... 1-cosxcos2xcos3xcos4x ... ∫xnexdx ... f x sin2xsin3x ...