If $\csc \frac{\pi}{32}+\csc \frac{\pi}{16}+\csc \frac{\pi}{8}+\csc \frac{\pi}{4}+\csc \frac{\pi}{2}=\cot \frac{\pi}{k}$.Find $k$
Let $\frac{\pi}{32}=\theta$
$\csc \frac{\pi}{32}+\csc \frac{\pi}{16}+\csc \frac{\pi}{8}+\csc \frac{\pi}{4}+\csc \frac{\pi}{2}=\frac{1}{\sin\theta}+\frac{1}{\sin2\theta}+\frac{1}{\sin4\theta}+\frac{1}{\sin8\theta}+\frac{1}{\sin16\theta}$
Now i am stuck.
$\endgroup$3 Answers
$\begingroup$HINT:
$$\csc2t+\cot2t=\dfrac{1+\cos2t}{\sin2t}=\cot t$$
$$\iff\csc2t=\cot t-\cot2t$$
Do you recognize the Telescoping nature by putting $2t=\dfrac\pi{32},\dfrac\pi{16},\dfrac\pi8,\dfrac\pi4, \dfrac\pi2$
$\endgroup$ $\begingroup$Hint: try to work out some smaller cases first, for example, $\csc\frac{\pi}{2}=\cot\frac{\pi}{k}$ or $\csc\frac{\pi}{4}+\csc\frac{\pi}{2}=\cot\frac{\pi}{k}$. Then try to generalize using induction.
$\endgroup$ 2 $\begingroup$We have the formula ie $\sin(2x)=2sinxcosx$ so we know value of $sin(45)$ substitute $45=2x$ so youll get $sin(22.5)=\pi/8$ . thus solving double angle identity many times you get all sines . then the work is almost done. To convert cos to sin use $sin^2(x)+cos^2(x)=1$
$\endgroup$
