\hspace{-16}$If $\mathbf{f(x)}$ is a defined function in such a way such that\\\\\\ $\mathbf{f(x+1)=\frac{f(x).\cos\left(\frac{\pi}{67}\right)+\sin\left(\frac{\pi}{67}\right)}{\cos\left(\frac{\pi}{67}\right)-f(x).\sin\left(\frac{\pi}{67}\right)}}$\\\\\\ If $\mathbf{f(0)=1}.$ Then find $\mathbf{f(2010)=}$
Ans = 1
-
UP 0 DOWN 0 0 1
1 Answers
341f(x+1) = \frac{f(x)+\tan \frac{\pi}{67}}{1-f(x) \tan \frac{\pi}{67}}
f(x) = \tan \phi\Rightarrow f(x+1)= \tan \left(\phi+ \frac{\pi}{67}\right)\Rightarrow f(x+n) = \tan \left(\phi+ \frac{n\pi}{67}\right)
by induction
thus f is periodic with period 67 and since 2010 = 67X30, f(2010+x) = f(x)
So f(2010)=f(0)=1
