Question

If $f_n\left(x\right)=\frac{sinx}{cos3x}\frac{sin3x}{cos{3}^{2}x}+\frac{sin{3}^{2}x}{cos{3}^{3}x}+.....+\frac{sin{3}^{n-1}}{cos{3}^{n}x}Then$f_2$\left(\pi /4\right)$+$f_3$(\pi /4)=$

• A. $0$
• B. $1$
• C. $-1$
• D. $2$

Answer 1

The correct option is C $-1$

$f_2\frac{\pi }{4}+f_3\frac{\pi }{4}$

$\frac{sin\frac{\pi }{4}}{cos\left(\frac{3\pi }{4}\right)}+\frac{sin\frac{3\pi }{4}}{cos\left(\frac{9\pi }{4}\right)}$+ $\frac{sin\frac{\pi }{4}}{cos\left(\frac{3\pi }{4}\right)}+\frac{sin\frac{3\pi }{4}}{cos\left(\frac{9\pi }{4}\right)}+\frac{sin\frac{\pi }{4}}{cos\left(\frac{3\pi }{4}\right)}+\frac{sin\frac{9\pi }{4}}{cos\left(\frac{27\pi }{4}\right)}$

$|\frac{\left(\frac{1}{\sqrt{2}}\right)}{(-\frac{1}{\sqrt{2}})}+\frac{(-\frac{1}{\sqrt{2}})}{\left(\frac{1}{\sqrt{2}}\right)}|+|\frac{\left(\frac{1}{\sqrt{2}}\right)}{\left(\frac{-1}{\sqrt{2}}\right)}+\frac{\left(\frac{1}{\sqrt{2}}\right)}{\left(\frac{1}{\sqrt{2}}\right)}+\frac{\left(\frac{1}{\sqrt{2}}\right)}{\left(\frac{-1}{\sqrt{2}}\right)}|$

$(-1+1)+(-1+1-1)$

$=-1$