Tunjukkan bahwa persamaan sin x1−cos x=1+cos xsin x dipenuhi pada semua x kecuali kelipatan π.

Pertanyaan

Tunjukkan bahwa persamaan $fraction numerator 1 minus cos space x over denominator sin space x end fraction equals fraction numerator sin space x over denominator 1 plus cos space x end fraction$ dipenuhi pada semua $x$ kecuali kelipatan $\mathrm\pi$ .

A. Acfreelance

Master Teacher

Jawaban terverifikasi

Jawaban

berdasarkan hitungan di atas terbukti bahwa persamaan $fraction numerator 1 minus cos space x over denominator sin space x end fraction equals fraction numerator sin space x over denominator 1 plus cos space x end fraction$ dipenuhi pada semua

kecuali kelipatan straight pi

Pembahasan

Diketahui $fraction numerator 1 minus cos space x over denominator sin space x end fraction$ maka diperoleh sebagai berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 1 minus cos space x over denominator sin space x end fraction end cell equals cell fraction numerator 1 minus cos space x over denominator sin space x end fraction cross times 1 end cell row blank equals cell fraction numerator 1 minus cos space x over denominator sin space x end fraction cross times fraction numerator 1 plus cos space x over denominator 1 plus cos space x end fraction end cell row blank equals cell fraction numerator 1 plus cos space x minus cos space x minus cos squared x over denominator sin space x open parentheses 1 plus cos space x close parentheses end fraction end cell row blank equals cell fraction numerator 1 minus cos squared x over denominator sin space x open parentheses 1 plus cos space x close parentheses end fraction end cell end table$

Berdasarkan identitas trigonometri yaitu sin squared x plus cos squared x equals 1 dengan sin squared x equals 1 minus cos squared x maka

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 1 minus cos space x over denominator sin space x end fraction end cell equals cell fraction numerator 1 minus cos squared x over denominator sin space x open parentheses 1 plus cos space x close parentheses end fraction end cell row blank equals cell fraction numerator sin squared x over denominator sin space x open parentheses 1 plus cos space x close parentheses end fraction end cell row blank equals cell fraction numerator up diagonal strike sin space x end strike times sin space x over denominator up diagonal strike sin space x end strike open parentheses 1 plus cos space x close parentheses end fraction end cell row blank equals cell fraction numerator sin space x over denominator 1 plus cos space x end fraction end cell end table$

Jadi berdasarkan hitungan di atas terbukti bahwa $fraction numerator 1 minus cos space x over denominator sin space x end fraction equals fraction numerator sin space x over denominator 1 plus cos space x end fraction$ agar persamaan $fraction numerator 1 minus cos space x over denominator sin space x end fraction equals fraction numerator sin space x over denominator 1 plus cos space x end fraction$ terpenuhi maka sin space x not equal to 0 sehingga

Jadi berdasarkan hitungan di atas terbukti bahwa persamaan $fraction numerator 1 minus cos space x over denominator sin space x end fraction equals fraction numerator sin space x over denominator 1 plus cos space x end fraction$ dipenuhi pada semua kecuali kelipatan straight pi .