Chebyshev Coefficients, Guessed

★ 6/10trigonometryalgebraPolynomialsTrigonometric Identities

Problem

Observe:

cos⁡2α=2cos⁡2α−1,\cos 2\alpha = 2\cos^2\alpha - 1, cos⁡4α=8cos⁡4α−8cos⁡2α+1,\cos 4\alpha = 8\cos^4\alpha - 8\cos^2\alpha + 1, cos⁡6α=32cos⁡6α−48cos⁡4α+18cos⁡2α−1,\cos 6\alpha = 32\cos^6\alpha - 48\cos^4\alpha + 18\cos^2\alpha - 1, cos⁡8α=128cos⁡8α−256cos⁡6α+160cos⁡4α−32cos⁡2α+1,\cos 8\alpha = 128\cos^8\alpha - 256\cos^6\alpha + 160\cos^4\alpha - 32\cos^2\alpha + 1, cos⁡10α=mcos⁡10α−1280cos⁡8α+1120cos⁡6α+ncos⁡4α+pcos⁡2α−1.\cos 10\alpha = m\cos^{10}\alpha - 1280\cos^8\alpha + 1120\cos^6\alpha + n\cos^4\alpha + p\cos^2\alpha - 1.

Infer m−n+pm - n + p.

Answer

Solution

Difficulty6/10
Topicstrigonometry, algebra, Polynomials, Trigonometric Identities

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