Demostrar las siguientes identidades trigonométricas (proceso, en fotos si puede):
1. (Cosx/coscx) + (cosx/sec-1) = 2tanx
2. (Sen^2x-tan^2x)/coscx= Senx
3. Senx/1-cosx =cscx
4. Tanx+cotx=secx*cscx
5. sen^2x+sen^2x*tan^2x= tan^2x
6. 1/cosec^2x + 1/sec^2x= 1
marianitamorena:
Demosta
Respuestas a la pregunta
Contestado por
1
El primero me parece que hay un error,
el segundo también...
Para el tercero:
![\frac{sin(x)}{1-cos(x)} =csc(x) \frac{sin(x)}{1-cos(x)} =csc(x)](https://tex.z-dn.net/?f=+%5Cfrac%7Bsin%28x%29%7D%7B1-cos%28x%29%7D+%3Dcsc%28x%29)
![\frac{sin(x)}{1-cos(x)}( \frac{1+cos(x)}{1+cos(x)} )= \frac{sin(x)(1+cos(x))}{1-cos ^{2}(x) } = \frac{sin(x)(1+cos(x))}{sin ^{2}(x) } = \frac{1+cos(x)}{sin(x)} =... \\ \\ ...= \frac{1}{sin(x)} + \frac{cos(x)}{sin(x)} =csc(x)+ctg(x) \frac{sin(x)}{1-cos(x)}( \frac{1+cos(x)}{1+cos(x)} )= \frac{sin(x)(1+cos(x))}{1-cos ^{2}(x) } = \frac{sin(x)(1+cos(x))}{sin ^{2}(x) } = \frac{1+cos(x)}{sin(x)} =... \\ \\ ...= \frac{1}{sin(x)} + \frac{cos(x)}{sin(x)} =csc(x)+ctg(x)](https://tex.z-dn.net/?f=%5Cfrac%7Bsin%28x%29%7D%7B1-cos%28x%29%7D%28+%5Cfrac%7B1%2Bcos%28x%29%7D%7B1%2Bcos%28x%29%7D+%29%3D+%5Cfrac%7Bsin%28x%29%281%2Bcos%28x%29%29%7D%7B1-cos+%5E%7B2%7D%28x%29+%7D+%3D+%5Cfrac%7Bsin%28x%29%281%2Bcos%28x%29%29%7D%7Bsin+%5E%7B2%7D%28x%29+%7D+%3D+%5Cfrac%7B1%2Bcos%28x%29%7D%7Bsin%28x%29%7D+%3D...+%5C%5C++%5C%5C+...%3D+%5Cfrac%7B1%7D%7Bsin%28x%29%7D+%2B+%5Cfrac%7Bcos%28x%29%7D%7Bsin%28x%29%7D+%3Dcsc%28x%29%2Bctg%28x%29)
cuarta:
![tan(x)+ctg(x)=sec(x)csc(x) \\ \\ \frac{sin(x)}{cos(x)} + \frac{cos(x)}{sin(x)} \\ \\ \frac{sin^{2}(x)+cos ^{2}(x) }{sin(x)cos(x)} \\ \\ \frac{1}{sin(x)cos(x)} = (\frac{1}{sin(x)} )( \frac{1}{cos(x)} )=csc(x)sec(x) tan(x)+ctg(x)=sec(x)csc(x) \\ \\ \frac{sin(x)}{cos(x)} + \frac{cos(x)}{sin(x)} \\ \\ \frac{sin^{2}(x)+cos ^{2}(x) }{sin(x)cos(x)} \\ \\ \frac{1}{sin(x)cos(x)} = (\frac{1}{sin(x)} )( \frac{1}{cos(x)} )=csc(x)sec(x)](https://tex.z-dn.net/?f=tan%28x%29%2Bctg%28x%29%3Dsec%28x%29csc%28x%29+%5C%5C++%5C%5C++%5Cfrac%7Bsin%28x%29%7D%7Bcos%28x%29%7D+%2B+%5Cfrac%7Bcos%28x%29%7D%7Bsin%28x%29%7D++%5C%5C++%5C%5C++%5Cfrac%7Bsin%5E%7B2%7D%28x%29%2Bcos+%5E%7B2%7D%28x%29++%7D%7Bsin%28x%29cos%28x%29%7D++%5C%5C++%5C%5C++%5Cfrac%7B1%7D%7Bsin%28x%29cos%28x%29%7D+%3D+%28%5Cfrac%7B1%7D%7Bsin%28x%29%7D+%29%28+%5Cfrac%7B1%7D%7Bcos%28x%29%7D+%29%3Dcsc%28x%29sec%28x%29)
quinta:
![sin^{2} (x)+(sin^{2} (x))tan ^{2} (x)=tan ^{2} (x) \\ \\ sin^{2} (x)(1+tan^{2}(x) ) \\ sin^{2} (x)(sec ^{2}(x) ) \\ sin ^{2} (x)( \frac{1}{cos ^{2}(x) } ) \\ \frac{sin^{2}(x) }{cos^{2}(x) } =tan ^{2} (x) sin^{2} (x)+(sin^{2} (x))tan ^{2} (x)=tan ^{2} (x) \\ \\ sin^{2} (x)(1+tan^{2}(x) ) \\ sin^{2} (x)(sec ^{2}(x) ) \\ sin ^{2} (x)( \frac{1}{cos ^{2}(x) } ) \\ \frac{sin^{2}(x) }{cos^{2}(x) } =tan ^{2} (x)](https://tex.z-dn.net/?f=sin%5E%7B2%7D+%28x%29%2B%28sin%5E%7B2%7D+%28x%29%29tan+%5E%7B2%7D+%28x%29%3Dtan+%5E%7B2%7D+%28x%29+%5C%5C++%5C%5C+sin%5E%7B2%7D+%28x%29%281%2Btan%5E%7B2%7D%28x%29+%29+%5C%5C+sin%5E%7B2%7D+%28x%29%28sec+%5E%7B2%7D%28x%29+%29+%5C%5C+sin+%5E%7B2%7D+%28x%29%28+%5Cfrac%7B1%7D%7Bcos+%5E%7B2%7D%28x%29+%7D+%29+%5C%5C++%5Cfrac%7Bsin%5E%7B2%7D%28x%29+%7D%7Bcos%5E%7B2%7D%28x%29+%7D+%3Dtan+%5E%7B2%7D+%28x%29)
sexta:
![\frac{1}{csc ^{2}(x) } + \frac{1}{sec^{2}(x) } =1 \\ \\ \frac{1}{ \frac{1}{sen^{2}(x) } } +\frac{1}{ \frac{1}{cos^{2}(x) } } = sin ^{2} (x)+ cos^{2}(x) =1 \frac{1}{csc ^{2}(x) } + \frac{1}{sec^{2}(x) } =1 \\ \\ \frac{1}{ \frac{1}{sen^{2}(x) } } +\frac{1}{ \frac{1}{cos^{2}(x) } } = sin ^{2} (x)+ cos^{2}(x) =1](https://tex.z-dn.net/?f=+%5Cfrac%7B1%7D%7Bcsc+%5E%7B2%7D%28x%29+%7D+%2B+%5Cfrac%7B1%7D%7Bsec%5E%7B2%7D%28x%29+%7D+%3D1+%5C%5C++%5C%5C++%5Cfrac%7B1%7D%7B+%5Cfrac%7B1%7D%7Bsen%5E%7B2%7D%28x%29+%7D+%7D+%2B%5Cfrac%7B1%7D%7B+%5Cfrac%7B1%7D%7Bcos%5E%7B2%7D%28x%29+%7D+%7D+%3D+sin+%5E%7B2%7D+%28x%29%2B+cos%5E%7B2%7D%28x%29+%3D1)
el segundo también...
Para el tercero:
cuarta:
quinta:
sexta:
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