Question

prove that 4sin3a.cos 3a + 4 cos 3a.sin 3a = 3 sin 4a

Answer 1

Answer on Question #43728-Engineering-Other

Prove that $4 \sin 3a \cos^3 a + 4 \cos 3a \sin^3 a = 3 \sin 4a$

Solution

We know that

$$\sin 2a = 2 \sin a \cos a, \quad \cos 2a = \cos^2 a - \sin^2 a$$ $$\sin 3a = 3 \sin a - 4 \sin^3 a, \quad \cos 3a = 4 \cos^3 a - 3 \cos a.$$

So

$$\begin{array}{l} 4 \sin 3a \cos^3 a + 4 \cos 3a \sin^3 a = 4 (3 \sin a - 4 \sin^3 a) \cos^3 a + 4 (4 \cos^3 a - 3 \cos a) \sin^3 a \\ = 12 \sin a \cos^3 a - 16 \sin^3 a \cos^3 a + 16 \sin^3 a \cos^3 a - 12 \cos a \sin^3 a \\ = 12 (\sin a \cos^3 a - \cos a \sin^3 a) = 6 \cdot (2 \sin a \cos a) (\cos^2 a - \sin^2 a) = 6 \sin 2a \cos 2a \\ = 3 \cdot (2 \sin 2a \cos 2a) = 3 \sin 4a. \end{array}$$

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