Question

Rewrite the expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1.

cos3x
Question options:

cos x - cos 3x

cos x + cos 3x

cos x - cos 3x - cos 2x

cos x + cos 3x + cos 2x

Answer 1

Answer on Question #75382 – Math – Trigonometry

$$\begin{array}{l} \cos x - \cos 3x = \cos x - 4\cos^3 x - 3\cos x = 4\cos x - 4\cos^3 x = 4\cos x(1 - \cos^2 x) = 4\cos x * \sin^2 \\ = 2\sin x * 2\sin x\cos x = 2\sin x * \sin^2 x \end{array}$$ $$\begin{array}{l} \cos x + \cos 3x = 4\cos^3 x - 2\cos x = 2\cos x(\cos^2 x - 1) = 2\cos x * (-\sin^2 x) = -2\cos x * \sin^2 x \\ = -\sin x * \sin^2 x \end{array}$$ $$\begin{array}{l} \cos x - \cos 3x - \cos 2x = \cos x - 4\cos^3 x + 3\cos x - \cos 2x = 4\cos x(1 - \cos^2 x) - \cos 2x \\ = 4\cos x * \sin^2 x - \cos 2x = 2\cos x(1 - \cos^2 x) - \cos 2x \\ = 2\cos x - 2\cos x * \cos^2 x - \cos^2 x \end{array}$$ $$\begin{array}{l} \cos x + \cos 3x + \cos 2x = \cos x + 4\cos^3 x - 3\cos x + \cos 2x = 4\cos^3 x - 2\cos x + \cos 2x \\ = 2\cos x(2\cos^2 x - 1) + \cos 2x = 2\cos x(1 + \cos^2 x - 1) + \cos 2x \\ = 2\cos x * \cos^2 x + \cos^2 x = \cos^2 x(2\cos x + 1) \end{array}$$

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