3.5 Multiple and Sub-multiple Angles

Use Sum-to-Product formulas (introduced in next section) or expand triple angles. Using identities: Numerator: $\sin x + (3\sin x - 4\sin^3 x) = 4\sin x - 4\sin^3 x$ Denominator: $\cos x + (4\cos^3 x - 3\cos x) = 4\cos^3 x - 2\cos x$ ...Better Method: Factorization Formulas. $\sin A + \sin B = 2 \sin \frac{A+B}{2} \cos \frac{A-B}{2}$ Num: $2 \sin 2x \cos(-x) = 2 \sin 2x \cos x$ Den: $2 \cos 2x \cos(-x) = 2 \cos 2x \cos x$ Ratio: $\frac{2 \sin 2x \cos x}{2 \cos 2x \cos x} = \tan 2x$.