Pertanyaan

Tunjukkan kebenaran identitas trigonometri berikut. j. 1 + cos x sin x = cosec x − cotan x

Tunjukkan kebenaran identitas trigonometri berikut.

j. $\frac{sin x}{1 + cos x} = cosec x - cotan x$

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

terbukti bahwa .

terbukti bahwa $fraction numerator sin space x over denominator 1 plus cos space x end fraction equals cosec space x minus cotan space x$ .

Pembahasan

Ingat! Akan ditunjukkan bahwa Perhatikan perhitungan berikut ini Dengan demikian terbukti bahwa .

Ingat!

$cotan space x equals fraction numerator cos space x over denominator sin space x end fraction$
$cosec space x equals fraction numerator 1 over denominator sin space x end fraction$

Akan ditunjukkan bahwa

$fraction numerator sin space x over denominator 1 plus cos space x end fraction equals cosec space x minus cotan space x$

Perhatikan perhitungan berikut ini

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

Dengan demikian terbukti bahwa $fraction numerator sin space x over denominator 1 plus cos space x end fraction equals cosec space x minus cotan space x$ .