MathLabs

Problem 4

Find all real solutions of cos⁡2x+cos⁡22x+cos⁡23x=1\cos^2 x+\cos^2 2x+\cos^2 3x=1.
Step 1 of 5: Replace the trigonometric multiples by a single variable
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c=cos⁡x,cos⁡2x=2c2−1,cos⁡3x=4c3−3cc=\cos x,\qquad \cos2x=2c^2-1,\qquad\cos3x=4c^3-3c
Detailed analysis

Let c=cos⁡xc=\cos x. The standard identities give cos⁡2x=2c2−1\cos2x=2c^2-1 and cos⁡3x=4c3−3c\cos3x=4c^3-3c.