I am a beginner at trigonometry, I want to know the answer to this question.
If $m^2\cos{\frac{2\pi}{15}}\cos{\frac{4\pi}{15}}\cos{\frac{8\pi}{15}}\cos{\frac{14\pi}{15}} = n^2$, then find the value of $\frac{m^2 – n^2}{n^2}$.
These are the steps I have tried and got stuck in the middle.
$$m^2\cos{\frac{2\pi}{15}}\cos{\frac{4\pi}{15}}\cos{\frac{7\pi}{15}}\cos{\frac{\pi}{15}} = n^2 $$
$$m^2(\frac{1}{4})(2\cos{\frac{2\pi}{15}}\cos{\frac{\pi}{15}})(2\cos{\frac{4\pi}{15}}\cos{\frac{7\pi}{15}}) = n^2$$
$$m^2(\frac{1}{4})(\cos{\frac{3\pi}{15}} + \cos{\frac{\pi}{15}})(\cos{\frac{11\pi}{15}} + \cos{\frac{3\pi}{15}}) = n^2$$
Fro here onwards I could not continue. These steps may be wrong so please check if my method of solving is correct and help me solve the question. Thanks:)
Best Answer
Hint:
$\cos\dfrac{14\pi}{15}=\cos\left(\pi-\dfrac\pi{15}\right)=?$
Use $\sin2x=2\sin x\cos x\iff\cos x=\dfrac{\sin2x}{2\sin x}$ repeatedly
Finally here $\sin\dfrac{16\pi}{15}=\sin\left(\pi+\dfrac\pi{15}\right)=?$