Question

¿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.

Answer 1

Recuerda que 
                     $\cot \alpha = \dfrac{1}{\tan \alpha}$

entonces
           $E=\dfrac{\cot(45-y)}{\cot(45+y)}=\dfrac{\tan(45+y)}{\tan(45-y)}\\ \\ \\ E=\dfrac{\dfrac{\tan45+\tan y}{1-\tan45\tan y}}{\dfrac{\tan45-\tan y}{1+\tan45\tan y}}\\ \\ \\ E=\dfrac{\dfrac{1+\tan y}{1-\tan y}}{\dfrac{1-\tan y}{1+\tan y}}$

           $E=\left(\dfrac{1+\tan y}{1-\tan y}\right)^2\\ \\ \\ E=\left(\dfrac{1+\dfrac{\sin y}{\cos y}}{1-\dfrac{\sin y}{\cos y}}\right)^2\\ \\ \\ E=\left(\dfrac{\cos y+\sin y}{\cos y-\sin y}\right)^2$

           $E=\dfrac{\cos^2y+2\sin y\cos y+\sin^2y}{\cos^2y-2\sin y\cos y+\sin^2y}\\ \\ \\ E=\dfrac{(\cos^2y+\sin^2y)+2\sin y\cos y}{(\cos^2y+\sin^2y)-2\sin y\cos y}\\ \\ \\ \boxed{E=\dfrac{1+2\sin y\cos y}{1-2\sin y\cos y}}$