trigonometry - If $frac{3-tan^2fracpi7}{1-tan^2fracpi7}=kcosfracpi7$, find $k$

I need help solving this question: $$\text{If }\frac{3-\tan^2\frac\pi7}{1-\tan^2\frac\pi7}=k\cos\frac\pi7\text{, find k.}$$ I simplified this down to:

$$\frac{4\cos^2\frac\pi7-1}{2\cos^2\frac\pi7-1}$$

But am unable to proceed further. The value of k is given to be 4, but I am unable to derive that result.

Kindly provide me with some insight, or with a step-by-step solution.

Thanks in advance,

Abhigyan

Answer

$$\begin{align} k&=\frac{4\cos^2\frac\pi7-1}{\cos\frac{\pi}{7}(2\cos^2\frac\pi7-1)}\\ &=\frac{2\cos\frac{2\pi}{7}+1}{\cos\frac{2\pi}{7}\cos\frac{\pi}{7}}\\ &=\frac{2\cos\frac{2\pi}{7}+1}{\frac{1}{2}(\cos\frac{3\pi}{7}+\cos\frac{\pi}{7})}\\ &=\frac{4\sin\frac{\pi}{7}\cos\frac{2\pi}{7}+2\sin\frac{\pi}{7}}{\sin\frac{\pi}{7}\cos\frac{3\pi}{7}+\sin\frac{\pi}{7}\cos\frac{\pi}{7}}\\ &=\frac{2\sin\frac{3\pi}{7}-2\sin\frac{\pi}{7}+2\sin\frac{\pi}{7}}{\frac{1}{2}(\sin\frac{4\pi}{7}-\sin\frac{2\pi}{7}+\sin\frac{2\pi}{7})}\\ &=\frac{4\sin\frac{3\pi}{7}}{\sin\frac{4\pi}{7}}\\ &=4 \end{align}$$