4. Tunjukkan kebenaran identitas trigonometri berikut. c. cosec A−cotan A1=cosec A+cotan A

Pertanyaan

4. Tunjukkan kebenaran identitas trigonometri berikut.

c. $fraction numerator 1 over denominator cosec space A minus cotan space A end fraction equals cosec space A plus cotan space A$

P. Tessalonika

Master Teacher

Mahasiswa/Alumni Universitas Negeri Medan

Jawaban terverifikasi

Pembahasan

Untuk membuktikan persamaan tersebut, kita akan menyederhanakan persamaan di ruas kiri untuk mendapatkan persamaan di ruas kanan, kemudian ingat kembali identitas trigonometri berikut.

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

Sehingga diperoleh:

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 1 over denominator cosec space A minus cotan space A end fraction end cell equals cell cosec space A plus cotan space A end cell row cell fraction numerator 1 over denominator begin display style fraction numerator 1 over denominator sin space A end fraction end style minus begin display style fraction numerator cos space A over denominator sin space A end fraction end style end fraction end cell equals cell cosec space A plus cotan space A end cell row cell fraction numerator 1 over denominator begin display style fraction numerator 1 minus cos space A over denominator sin space A end fraction end style end fraction end cell equals cell cosec space A plus cotan space A end cell row cell fraction numerator sin space A over denominator 1 minus cos space A end fraction end cell equals cell cosec space A plus cotan space A end cell row cell fraction numerator sin space A over denominator 1 minus cos space A end fraction cross times fraction numerator 1 plus cos space A over denominator 1 plus cos space A end fraction end cell equals cell cosec space A plus cotan space A end cell row cell fraction numerator sin space A open parentheses 1 plus cos space A close parentheses over denominator 1 minus cos squared space A end fraction end cell equals cell cosec space A plus cotan space A end cell row cell fraction numerator sin space A open parentheses 1 plus cos space A close parentheses over denominator 1 minus open parentheses 1 minus sin squared space A close parentheses end fraction end cell equals cell cosec space A plus cotan space A end cell row cell fraction numerator up diagonal strike sin space A end strike open parentheses 1 plus cos space A close parentheses over denominator sin to the power of up diagonal strike 2 end exponent space A end fraction end cell equals cell cosec space A plus cotan space A end cell row cell fraction numerator 1 plus cos space A over denominator sin space A end fraction end cell equals cell cosec space A plus cotan space A end cell row cell fraction numerator 1 over denominator sin space A end fraction plus fraction numerator cos space A over denominator sin space A end fraction end cell equals cell cosec space A plus cotan space A end cell row cell cosec space A plus cotan space A end cell equals cell cosec space A plus cotan space A space begin bold style left parenthesis terbukti right parenthesis end style end cell row blank blank blank end table$

Dengan demikian, terbukti bahwa :

$fraction numerator 1 over denominator cosec space A minus cotan space A end fraction equals cosec space A plus cotan space A$