Question

If $f_n\left(x\right)=\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 3^{2}x}+\frac{\sin 3^{2}x}{\cos 3^{3}x}+\dots +\frac{\sin 3^{n-1}x}{\cos 3^{n}x}$ then $f_2\left(\frac{\pi }{4}\right)+f_3\left(\frac{\pi }{4}\right)=$

• A. 0
• B. 1
• C. -1
• D. 2

Answer 1

Answer: (C)

-1

$f_2\left(x\right)=\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 3^{2}x}$

$f_2\left(\pi /4\right)=\frac{\sin \left(\pi /4\right)}{\cos \left(3\pi /4\right)}+\frac{\sin \left(3\pi /4\right)}{\cos \left(9\pi /4\right)}$

$f_2\left(\pi /4\right)=f_2\left(\pi /4\right)+\frac{\sin \left(9\pi /4\right)}{\cos \left(27\pi /4\right)}$

$f_2\left(\pi /4\right)=\frac{\frac{1}{\sqrt{2}}}{\frac{-1}{\sqrt{2}}}+\frac{\left(\frac{+1}{\sqrt{2}}\right)}{\frac{1}{\sqrt{2}}}$

$=-1+1=0$

$f_3\left(\pi /4\right)=0+\frac{1/\sqrt{2}}{-1/\sqrt{2}}=-1$

$f_2\left(\pi /4\right)+f_3\left(\pi /4\right)=-1$.