4. Poenostavi izraz \( \frac{\sin 2 x-2 \sin x}{1+\cos 2 x-2 \cos x} \)
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Za poenostavitev izraza \( \frac{\sin 2x - 2 \sin x}{1 + \cos 2x - 2 \cos x} \), najprej uporabimo identitete za sinus in kosinus. Spomnimo se, da \( \sin 2x = 2 \sin x \cos x \) in \( \cos 2x = 2 \cos^2 x - 1 \). Tako lahko izraz preoblikujemo. V zgornjem delu: \[ \sin 2x - 2 \sin x = 2 \sin x \cos x - 2 \sin x = 2 \sin x (\cos x - 1). \] V spodnjem delu izraz preoblikujemo: \[ 1 + \cos 2x - 2 \cos x = 1 + (2 \cos^2 x - 1) - 2 \cos x = 2 \cos^2 x - 2 \cos x = 2 \cos x (\cos x - 1). \] Sedaj lahko izraz poenostavimo: \[ \frac{2 \sin x (\cos x - 1)}{2 \cos x (\cos x - 1)}. \] Obe strani razdelimo s \( 2 (\cos x - 1) \) (seveda, pod pogojem, da \( \cos x \neq 1 \)): \[ \frac{\sin x}{\cos x} = \tan x. \] Tako je poenostavljeni izraz: \[ \tan x. \]