Answer
See the steps.
Work Step by Step
$\sin{5x} + \sin{3x} = 2 \sin{4x} \cos{x}$
$\cos{3x} + \cos{5x} = 2 \cos{4x} \cos{-x} = 2 \cos{4x} \cos{x}$
$RHS = \dfrac{2 \sin{4x} \cos{x}}{2 \cos{4x} \cos{x}} = \tan{4x} = LHS$
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