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Question Number 207518 by hardmath last updated on 17/May/24

$$\cos \mathbf{x}\cos 3\mathbf{x}=\cos 5\mathbf{x}\cos 7\mathbf{x}$$$$\Rightarrow \mathbf{x}=?$$

Answered by Frix last updated on 17/May/24

$$\cos x\cos 3x=\cos 5x\cos 7x$$$$\frac{\cos 2x+\cos 4x}{2}=\frac{\cos 2x+\cos 12x}{2}$$$$\cos 4x=\cos 12x$$$$\cos y=\cos 3y$$$$\cos y=(1-{4\sin }^{2}y)\cos y$$$$\cos y-(1-{4\sin }^{2}y)\cos y=0$$$$4\cos y{\sin }^{2}y=0$$$$\cos y=0\vee \sin y=0$$$$\cos 4x=0\vee \sin 4x=0$$$$x=\frac{(2n+1)\pi }{8}\vee x=\frac{2n\pi }{8}$$$$x=\frac{n\pi }{8}$$

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