Soal Identitas Trigonometri Tersulit di Dunia Beserta Penyelesaiannya

Mungkin layak soal ini mendapatkan ‘gelar’ tersebut. Soal ini saya peroleh dari Yahoo Answers Amerika Serikat setahun yang lalu dari seorang dosen matematika. Sayangnya saya lupa tautan (link) nya. Saya bilang luar biasa sang pembuat soal ini! Soal tersebut adalah (saya modifikasi terjemahannya dalam bahasa Indonesia) karena saya lupa bunyinya, Buktikan bahwa:

\displaystyle \frac{4\cos^4 x-4\sin x\cos^3 x-2\sin^2 x\tan x-\tan^2 x-2\cos 2x}{\sec^2x+\cos 2x}=\cos 2x-\sin 2x

 

Jawab:

\displaystyle \frac{4\cos^4 x-4\sin x\cos^3 x-2\sin^2 x\tan x-\tan^2 x-2\cos 2x}{\sec^2x+\cos 2x}\\= \frac{2\cos^2 x}{2\cos^2 x}~\frac{4\cos^4 x-4\sin x\cos^3 x-2\sin^2 x\frac{\sin x}{\cos x}-\frac{\sin^2 x}{\cos^2 x}-2\cos 2x}{\frac{1}{\cos^2 x}+\cos 2x}\\ =\frac{8\cos^6 x-8\sin x\cos^5 x-4\sin^3 x\cos x-2\sin^2 x-4\cos 2x\cos^2 x}{2+2\cos^2 x\cos 2x}\\ =\frac{(2\cos^2 x)^3-(2\sin x\cos x)(2\cos^2 x)^2-(2\sin x\cos x)(2\sin^2 x)-2\sin^2 x-(2\cos 2x)(2\cos^2 x)}{2+(\cos 2x)(2\cos^2 x)}\\ =\frac{(1+\cos 2x)^3-(\sin 2x)(1+\cos 2x)^2-(\sin 2x)(1-\cos 2x)-(1-\cos 2x)-(2\cos 2x)(1+\cos 2x)}{2+\cos 2x(1+\cos 2x)}\\=\frac{(1+3\cos 2x+3\cos^2 x+\cos^3 2x)-(\sin 2x)(1+2\cos 2x+\cos^2 x)-(\sin 2x)(1-\cos 2x)-(1-\cos 2x)-(2\cos 2x)(1+\cos 2x)}{2+\cos 2x+\cos^2 x}\\ =\frac{1+3\cos 2x+3\cos^2 x+\cos^3 2x-\sin 2x-2\sin 2x\cos 2x-\sin 2x\cos^2 x-\sin 2x+\sin 2x\cos 2x-1+\cos 2x-2\cos 2x-2\cos^2 2x}{2+\cos 2x+\cos^2 x}\\ =\frac{2\cos 2x+\cos^2 2x+\cos^3 2x-2\sin 2x-\sin 2x\cos 2x-\sin 2x\cos^2 2x}{2+\cos 2x+\cos^2 x}\\ =\frac{\cos 2x(2+\cos 2x+\cos^2 2x)-\sin 2x(2+\cos 2x+\cos^2 2x)}{2+\cos 2x+\cos^2 x}\\ =\frac{(\cos 2x-\sin 2x)(2+\cos 2x+\cos^2 2x)}{(2+\cos 2x+\cos^2 2x)}\\ =\cos 2x-\sin 2x
Terbukti!

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55 pemikiran pada “Soal Identitas Trigonometri Tersulit di Dunia Beserta Penyelesaiannya

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