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}
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