Problem (5) : Trigonometri


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Jika f(x) = \frac{sec \quad 2x \sqrt{1-cos \quad 4x}}{cos^4 \quad x(1-tan^4 \quad x)}

Buktikan f(x) = \sqrt{2} sec 2x tan 2x

Penyelesaian :

f(x) = \frac{sec \quad 2x \sqrt{1-cos \quad 4x}}{cos^4 \quad x(1-tan^4 \quad x)}

= \frac{sec \quad 2x \sqrt{2 \quad sin^2 \quad 2x}}{cos^4 \quad x(1-\frac{sin^4 \quad x}{cos^4 \quad x})} (Sifat-Sifat Dasar Trigonometri)

= \frac{1}{cos^4 \quad x(\frac{cos^4 \quad x-sin^4 \quad x}{cos^4 \quad x})} sec 2x \sqrt{2} sin 2x

= \frac{1}{cos^4 \quad x-sin^4 \quad x} \sqrt{2} sec 2x sin 2x

= \frac{1}{(cos^2 \quad x+sin^2 \quad x)(cos^2 \quad x-sin^2 \quad x)} \sqrt{2} sec 2x sin 2x

= \frac{sin \quad 2x}{(1)(cos \quad 2x)} \sqrt{2} sec 2x

= \sqrt{2} sec 2x tan 2x

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