¿Ayuda con este ejercicio de identidades trigonométricas?
Demostrar que:
cot(45-y)/cot(45+y)=1+2sinycosy/1-2sinycosy
Por favor, me serviría si explican paso a paso lo que hacen.
Respuestas
Respuesta dada por:
1
Recuerda que
![\cot \alpha = \dfrac{1}{\tan \alpha} \cot \alpha = \dfrac{1}{\tan \alpha}](https://tex.z-dn.net/?f=%5Ccot+%5Calpha+%3D+%5Cdfrac%7B1%7D%7B%5Ctan+%5Calpha%7D)
entonces
![E=\dfrac{\cot(45-y)}{\cot(45+y)}=\dfrac{\tan(45+y)}{\tan(45-y)}\\ \\ \\
E=\dfrac{\dfrac{\tan45+\tan y}{1-\tan45\tan y}}{\dfrac{\tan45-\tan y}{1+\tan45\tan y}}\\ \\ \\
E=\dfrac{\dfrac{1+\tan y}{1-\tan y}}{\dfrac{1-\tan y}{1+\tan y}}\\ \\ \\
E=\dfrac{\cot(45-y)}{\cot(45+y)}=\dfrac{\tan(45+y)}{\tan(45-y)}\\ \\ \\
E=\dfrac{\dfrac{\tan45+\tan y}{1-\tan45\tan y}}{\dfrac{\tan45-\tan y}{1+\tan45\tan y}}\\ \\ \\
E=\dfrac{\dfrac{1+\tan y}{1-\tan y}}{\dfrac{1-\tan y}{1+\tan y}}\\ \\ \\](https://tex.z-dn.net/?f=E%3D%5Cdfrac%7B%5Ccot%2845-y%29%7D%7B%5Ccot%2845%2By%29%7D%3D%5Cdfrac%7B%5Ctan%2845%2By%29%7D%7B%5Ctan%2845-y%29%7D%5C%5C+%5C%5C+%5C%5C%0AE%3D%5Cdfrac%7B%5Cdfrac%7B%5Ctan45%2B%5Ctan+y%7D%7B1-%5Ctan45%5Ctan+y%7D%7D%7B%5Cdfrac%7B%5Ctan45-%5Ctan+y%7D%7B1%2B%5Ctan45%5Ctan+y%7D%7D%5C%5C+%5C%5C+%5C%5C%0AE%3D%5Cdfrac%7B%5Cdfrac%7B1%2B%5Ctan+y%7D%7B1-%5Ctan+y%7D%7D%7B%5Cdfrac%7B1-%5Ctan+y%7D%7B1%2B%5Ctan+y%7D%7D%5C%5C+%5C%5C+%5C%5C%0A)
![E=\left(\dfrac{1+\tan y}{1-\tan y}\right)^2\\ \\ \\
E=\left(\dfrac{1+\dfrac{\sin y}{\cos y}}{1-\dfrac{\sin y}{\cos y}}\right)^2\\ \\ \\
E=\left(\dfrac{\cos y+\sin y}{\cos y-\sin y}\right)^2\\ \\ \\
E=\left(\dfrac{1+\tan y}{1-\tan y}\right)^2\\ \\ \\
E=\left(\dfrac{1+\dfrac{\sin y}{\cos y}}{1-\dfrac{\sin y}{\cos y}}\right)^2\\ \\ \\
E=\left(\dfrac{\cos y+\sin y}{\cos y-\sin y}\right)^2\\ \\ \\](https://tex.z-dn.net/?f=E%3D%5Cleft%28%5Cdfrac%7B1%2B%5Ctan+y%7D%7B1-%5Ctan+y%7D%5Cright%29%5E2%5C%5C+%5C%5C+%5C%5C%0AE%3D%5Cleft%28%5Cdfrac%7B1%2B%5Cdfrac%7B%5Csin+y%7D%7B%5Ccos+y%7D%7D%7B1-%5Cdfrac%7B%5Csin+y%7D%7B%5Ccos+y%7D%7D%5Cright%29%5E2%5C%5C+%5C%5C+%5C%5C%0AE%3D%5Cleft%28%5Cdfrac%7B%5Ccos+y%2B%5Csin+y%7D%7B%5Ccos+y-%5Csin+y%7D%5Cright%29%5E2%5C%5C+%5C%5C+%5C%5C%0A)
![E=\dfrac{\cos^2y+2\sin y\cos y+\sin^2y}{\cos^2y-2\sin y\cos y+\sin^2y}\\ \\ \\
E=\dfrac{(\cos^2y+\sin^2y)+2\sin y\cos y}{(\cos^2y+\sin^2y)-2\sin y\cos y}\\ \\ \\
\boxed{E=\dfrac{1+2\sin y\cos y}{1-2\sin y\cos y}} E=\dfrac{\cos^2y+2\sin y\cos y+\sin^2y}{\cos^2y-2\sin y\cos y+\sin^2y}\\ \\ \\
E=\dfrac{(\cos^2y+\sin^2y)+2\sin y\cos y}{(\cos^2y+\sin^2y)-2\sin y\cos y}\\ \\ \\
\boxed{E=\dfrac{1+2\sin y\cos y}{1-2\sin y\cos y}}](https://tex.z-dn.net/?f=E%3D%5Cdfrac%7B%5Ccos%5E2y%2B2%5Csin+y%5Ccos+y%2B%5Csin%5E2y%7D%7B%5Ccos%5E2y-2%5Csin+y%5Ccos+y%2B%5Csin%5E2y%7D%5C%5C+%5C%5C+%5C%5C%0AE%3D%5Cdfrac%7B%28%5Ccos%5E2y%2B%5Csin%5E2y%29%2B2%5Csin+y%5Ccos+y%7D%7B%28%5Ccos%5E2y%2B%5Csin%5E2y%29-2%5Csin+y%5Ccos+y%7D%5C%5C+%5C%5C+%5C%5C%0A%5Cboxed%7BE%3D%5Cdfrac%7B1%2B2%5Csin+y%5Ccos+y%7D%7B1-2%5Csin+y%5Ccos+y%7D%7D)
entonces
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