Pertanyaan

Buktikan identitas berikut. b. tan α − cot α = sin α cos α 2 sin 2 α − 1

Buktikan identitas berikut.

b. $tan α - cot α = \frac{2 sin ^{2} α - 1}{sin α cos α}$

D. Enty

Master Teacher

Jawaban terverifikasi

Jawaban

terbukti bahwa .

terbukti bahwa $tan space alpha minus cot space alpha equals fraction numerator 2 space sin squared alpha minus 1 over denominator sin space alpha space cos space alpha end fraction$ .

Pembahasan

Diketahui identitas . Karena , maka Diketahui pula identitas , maka . Ruas kiri pada soal di atas dapat diubah menjadi Jadi, terbukti bahwa .

Diketahui identitas $tan space alpha equals fraction numerator sin space alpha over denominator cos space alpha end fraction$ .

Karena $cot space alpha equals fraction numerator 1 over denominator tan space alpha end fraction$ , maka

$table attributes columnalign right center left columnspacing 0px end attributes row cell cot space alpha end cell equals cell fraction numerator 1 over denominator begin display style fraction numerator sin space alpha over denominator cos space alpha end fraction end style end fraction end cell row blank equals cell 1 cross times fraction numerator cos space alpha over denominator sin space alpha end fraction end cell row blank equals cell fraction numerator cos space alpha over denominator sin space alpha end fraction end cell end table$

Diketahui pula identitas sin squared alpha plus cos squared alpha equals 1 , maka cos squared alpha equals 1 minus sin squared alpha .

Ruas kiri pada soal di atas dapat diubah menjadi

$table attributes columnalign right center left columnspacing 0px end attributes row cell tan space alpha minus cot space alpha end cell equals cell fraction numerator sin space alpha over denominator cos space alpha end fraction minus fraction numerator cos space alpha over denominator sin space alpha end fraction end cell row blank equals cell fraction numerator sin space alpha space left parenthesis sin space alpha right parenthesis over denominator cos space alpha space sin space alpha end fraction minus fraction numerator cos space alpha space left parenthesis cos space alpha right parenthesis over denominator sin space alpha space cos space alpha space end fraction end cell row blank equals cell fraction numerator sin squared space alpha minus cos squared alpha over denominator sin space alpha space cos space alpha space end fraction end cell row blank equals cell fraction numerator sin squared alpha minus left parenthesis 1 minus sin squared alpha right parenthesis over denominator sin space alpha space cos space alpha space end fraction end cell row blank equals cell fraction numerator sin squared alpha plus sin squared alpha minus 1 over denominator sin space alpha space cos space alpha space end fraction end cell row blank equals cell fraction numerator 2 space sin squared alpha minus 1 over denominator sin space alpha space cos space alpha space end fraction end cell row blank blank blank end table$

Jadi, terbukti bahwa $tan space alpha minus cot space alpha equals fraction numerator 2 space sin squared alpha minus 1 over denominator sin space alpha space cos space alpha end fraction$ .