Question

Prove that
$\frac{ \sin(5x) - 2 \sin(3x) + \sin(x) }{ \cos(5x) - \cos(x) } = \tan(x)$
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Answer 1

$\small\bf \colorbox{green}{Verified Answer}$

We use sin A + sin B = 2 sin (A+B)/2 Cos (A_B)/2

cos A - cos B = - 2 sin (A+B)/2 Sin(A-B)/2

Combine sin5x and sinx in the numerator. And combine cos5x and cosx in the denom.

$\begin{gathered}\frac{sin5x-2sin3x+sinx}{cos5x-cosx}=\\\\\frac{2sin3x\ cos2x-2sin3x}{-2sin(3x)sin(2x)}\\\\=\frac{1-cos2x}{sinx}\\\\=\frac{2sin^2x}{2sinxcosx}=tanx\end{gathered}$