Prakalkulus Contoh

\frac{tan (x) - cot (x)}{tan (x) + cot (x)} = 1 - 2 {cos}^{2} (x)

$\frac{\tan (x) - \cot (x)}{\tan (x) + \cot (x)} = 1 - 2 \cos^{2} (x)$

Langkah 1

Mulai dari sisi kiri.

\frac{tan (x) - cot (x)}{tan (x) + cot (x)}

$\frac{\tan (x) - \cot (x)}{\tan (x) + \cot (x)}$

Langkah 2

Sederhanakan pernyataannya.

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Langkah 2.1

Sederhanakan pembilangnya.

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Langkah 2.1.1

Tulis kembali

tan (x)

$\tan (x)$ dalam bentuk sinus dan kosinus.

\frac{\frac{sin (x)}{cos (x)} - cot (x)}{tan (x) + cot (x)}

$\frac{\frac{\sin (x)}{\cos (x)} - \cot (x)}{\tan (x) + \cot (x)}$

Langkah 2.1.2

Tulis kembali

cot (x)

$\cot (x)$ dalam bentuk sinus dan kosinus.

\frac{\frac{sin (x)}{cos (x)} - \frac{cos (x)}{sin (x)}}{tan (x) + cot (x)}

$\frac{\frac{\sin (x)}{\cos (x)} - \frac{\cos (x)}{\sin (x)}}{\tan (x) + \cot (x)}$

\frac{\frac{sin (x)}{cos (x)} - \frac{cos (x)}{sin (x)}}{tan (x) + cot (x)}

$\frac{\frac{\sin (x)}{\cos (x)} - \frac{\cos (x)}{\sin (x)}}{\tan (x) + \cot (x)}$

Langkah 2.2

Sederhanakan penyebutnya.

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Langkah 2.2.1

Tulis kembali

tan (x)

$\tan (x)$ dalam bentuk sinus dan kosinus.

\frac{\frac{sin (x)}{cos (x)} - \frac{cos (x)}{sin (x)}}{\frac{sin (x)}{cos (x)} + cot (x)}

$\frac{\frac{\sin (x)}{\cos (x)} - \frac{\cos (x)}{\sin (x)}}{\frac{\sin (x)}{\cos (x)} + \cot (x)}$

Langkah 2.2.2

Tulis kembali

cot (x)

$\cot (x)$ dalam bentuk sinus dan kosinus.

\frac{\frac{sin (x)}{cos (x)} - \frac{cos (x)}{sin (x)}}{\frac{sin (x)}{cos (x)} + \frac{cos (x)}{sin (x)}}

$\frac{\frac{\sin (x)}{\cos (x)} - \frac{\cos (x)}{\sin (x)}}{\frac{\sin (x)}{\cos (x)} + \frac{\cos (x)}{\sin (x)}}$

\frac{\frac{sin (x)}{cos (x)} - \frac{cos (x)}{sin (x)}}{\frac{sin (x)}{cos (x)} + \frac{cos (x)}{sin (x)}}

$\frac{\frac{\sin (x)}{\cos (x)} - \frac{\cos (x)}{\sin (x)}}{\frac{\sin (x)}{\cos (x)} + \frac{\cos (x)}{\sin (x)}}$

Langkah 2.3

Multiply the numerator and denominator of the fraction by

cos (x) sin (x)

$\cos (x) \sin (x)$ .

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Langkah 2.3.1

Kalikan

\frac{\frac{sin (x)}{cos (x)} - \frac{cos (x)}{sin (x)}}{\frac{sin (x)}{cos (x)} + \frac{cos (x)}{sin (x)}}

$\frac{\frac{\sin (x)}{\cos (x)} - \frac{\cos (x)}{\sin (x)}}{\frac{\sin (x)}{\cos (x)} + \frac{\cos (x)}{\sin (x)}}$ dengan

\frac{cos (x) sin (x)}{cos (x) sin (x)}

$\frac{\cos (x) \sin (x)}{\cos (x) \sin (x)}$ .

\frac{cos (x) sin (x)}{cos (x) sin (x)} \cdot \frac{\frac{sin (x)}{cos (x)} - \frac{cos (x)}{sin (x)}}{\frac{sin (x)}{cos (x)} + \frac{cos (x)}{sin (x)}}

$\frac{\cos (x) \sin (x)}{\cos (x) \sin (x)} \cdot \frac{\frac{\sin (x)}{\cos (x)} - \frac{\cos (x)}{\sin (x)}}{\frac{\sin (x)}{\cos (x)} + \frac{\cos (x)}{\sin (x)}}$

Langkah 2.3.2

Gabungkan.

\frac{cos (x) sin (x) (\frac{sin (x)}{cos (x)} - \frac{cos (x)}{sin (x)})}{cos (x) sin (x) (\frac{sin (x)}{cos (x)} + \frac{cos (x)}{sin (x)})}

$\frac{\cos (x) \sin (x) (\frac{\sin (x)}{\cos (x)} - \frac{\cos (x)}{\sin (x)})}{\cos (x) \sin (x) (\frac{\sin (x)}{\cos (x)} + \frac{\cos (x)}{\sin (x)})}$

\frac{cos (x) sin (x) (\frac{sin (x)}{cos (x)} - \frac{cos (x)}{sin (x)})}{cos (x) sin (x) (\frac{sin (x)}{cos (x)} + \frac{cos (x)}{sin (x)})}

$\frac{\cos (x) \sin (x) (\frac{\sin (x)}{\cos (x)} - \frac{\cos (x)}{\sin (x)})}{\cos (x) \sin (x) (\frac{\sin (x)}{\cos (x)} + \frac{\cos (x)}{\sin (x)})}$

Langkah 2.4

Terapkan sifat distributif.

\frac{cos (x) sin (x) \frac{sin (x)}{cos (x)} + cos (x) sin (x) (- \frac{cos (x)}{sin (x)})}{cos (x) sin (x) \frac{sin (x)}{cos (x)} + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) (- \frac{\cos (x)}{\sin (x)})}{\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5

Sederhanakan dengan cara membatalkan .

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Langkah 2.5.1

Batalkan faktor persekutuan dari

cos (x)

$\cos (x)$ .

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Langkah 2.5.1.1

Faktorkan

cos (x)

$\cos (x)$ dari

cos (x) sin (x)

$\cos (x) \sin (x)$ .

\frac{cos (x) (sin (x)) \frac{sin (x)}{cos (x)} + cos (x) sin (x) (- \frac{cos (x)}{sin (x)})}{cos (x) sin (x) \frac{sin (x)}{cos (x)} + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\cos (x) (\sin (x)) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) (- \frac{\cos (x)}{\sin (x)})}{\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.1.2

Batalkan faktor persekutuan.

\frac{cos (x) sin (x) \frac{sin (x)}{cos (x)} + cos (x) sin (x) (- \frac{cos (x)}{sin (x)})}{cos (x) sin (x) \frac{sin (x)}{cos (x)} + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) (- \frac{\cos (x)}{\sin (x)})}{\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.1.3

Tulis kembali pernyataannya.

\frac{sin (x) sin (x) + cos (x) sin (x) (- \frac{cos (x)}{sin (x)})}{cos (x) sin (x) \frac{sin (x)}{cos (x)} + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\sin (x) \sin (x) + \cos (x) \sin (x) (- \frac{\cos (x)}{\sin (x)})}{\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

\frac{sin (x) sin (x) + cos (x) sin (x) (- \frac{cos (x)}{sin (x)})}{cos (x) sin (x) \frac{sin (x)}{cos (x)} + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\sin (x) \sin (x) + \cos (x) \sin (x) (- \frac{\cos (x)}{\sin (x)})}{\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.2

Naikkan

sin (x)

$\sin (x)$ menjadi pangkat

1

$1$ .

\frac{{sin}^{1} (x) sin (x) + cos (x) sin (x) (- \frac{cos (x)}{sin (x)})}{cos (x) sin (x) \frac{sin (x)}{cos (x)} + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\sin^{1} (x) \sin (x) + \cos (x) \sin (x) (- \frac{\cos (x)}{\sin (x)})}{\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.3

Naikkan

sin (x)

$\sin (x)$ menjadi pangkat

1

$1$ .

\frac{{sin}^{1} (x) {sin}^{1} (x) + cos (x) sin (x) (- \frac{cos (x)}{sin (x)})}{cos (x) sin (x) \frac{sin (x)}{cos (x)} + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\sin^{1} (x) \sin^{1} (x) + \cos (x) \sin (x) (- \frac{\cos (x)}{\sin (x)})}{\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.4

Gunakan kaidah pangkat

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ untuk menggabungkan pangkat.

\frac{{sin (x)}^{1 + 1} + cos (x) sin (x) (- \frac{cos (x)}{sin (x)})}{cos (x) sin (x) \frac{sin (x)}{cos (x)} + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{{\sin (x)}^{1 + 1} + \cos (x) \sin (x) (- \frac{\cos (x)}{\sin (x)})}{\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.5

Tambahkan

1

$1$ dan

1

$1$ .

\frac{{sin}^{2} (x) + cos (x) sin (x) (- \frac{cos (x)}{sin (x)})}{cos (x) sin (x) \frac{sin (x)}{cos (x)} + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\sin^{2} (x) + \cos (x) \sin (x) (- \frac{\cos (x)}{\sin (x)})}{\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.6

Batalkan faktor persekutuan dari

sin (x)

$\sin (x)$ .

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Langkah 2.5.6.1

Pindahkan negatif pertama pada

- \frac{cos (x)}{sin (x)}

$- \frac{\cos (x)}{\sin (x)}$ ke dalam pembilangnya.

\frac{{sin}^{2} (x) + cos (x) sin (x) \frac{- cos (x)}{sin (x)}}{cos (x) sin (x) \frac{sin (x)}{cos (x)} + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\sin^{2} (x) + \cos (x) \sin (x) \frac{- \cos (x)}{\sin (x)}}{\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.6.2

Faktorkan

sin (x)

$\sin (x)$ dari

cos (x) sin (x)

$\cos (x) \sin (x)$ .

\frac{{sin}^{2} (x) + sin (x) cos (x) \frac{- cos (x)}{sin (x)}}{cos (x) sin (x) \frac{sin (x)}{cos (x)} + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\sin^{2} (x) + \sin (x) \cos (x) \frac{- \cos (x)}{\sin (x)}}{\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.6.3

Batalkan faktor persekutuan.

\frac{{sin}^{2} (x) + sin (x) cos (x) \frac{- cos (x)}{sin (x)}}{cos (x) sin (x) \frac{sin (x)}{cos (x)} + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\sin^{2} (x) + \sin (x) \cos (x) \frac{- \cos (x)}{\sin (x)}}{\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.6.4

Tulis kembali pernyataannya.

\frac{{sin}^{2} (x) + cos (x) (- cos (x))}{cos (x) sin (x) \frac{sin (x)}{cos (x)} + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\sin^{2} (x) + \cos (x) (- \cos (x))}{\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

\frac{{sin}^{2} (x) + cos (x) (- cos (x))}{cos (x) sin (x) \frac{sin (x)}{cos (x)} + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\sin^{2} (x) + \cos (x) (- \cos (x))}{\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.7

Naikkan

cos (x)

$\cos (x)$ menjadi pangkat

1

$1$ .

\frac{{sin}^{2} (x) - ({cos}^{1} (x) cos (x))}{cos (x) sin (x) \frac{sin (x)}{cos (x)} + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\sin^{2} (x) - (\cos^{1} (x) \cos (x))}{\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.8

Naikkan

cos (x)

$\cos (x)$ menjadi pangkat

1

$1$ .

\frac{{sin}^{2} (x) - ({cos}^{1} (x) {cos}^{1} (x))}{cos (x) sin (x) \frac{sin (x)}{cos (x)} + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\sin^{2} (x) - (\cos^{1} (x) \cos^{1} (x))}{\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.9

Gunakan kaidah pangkat

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ untuk menggabungkan pangkat.

\frac{{sin}^{2} (x) - {cos (x)}^{1 + 1}}{cos (x) sin (x) \frac{sin (x)}{cos (x)} + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\sin^{2} (x) - {\cos (x)}^{1 + 1}}{\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.10

Tambahkan

1

$1$ dan

1

$1$ .

\frac{{sin}^{2} (x) - {cos}^{2} (x)}{cos (x) sin (x) \frac{sin (x)}{cos (x)} + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\sin^{2} (x) - \cos^{2} (x)}{\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.11

Batalkan faktor persekutuan dari

cos (x)

$\cos (x)$ .

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Langkah 2.5.11.1

Faktorkan

cos (x)

$\cos (x)$ dari

cos (x) sin (x)

$\cos (x) \sin (x)$ .

\frac{{sin}^{2} (x) - {cos}^{2} (x)}{cos (x) (sin (x)) \frac{sin (x)}{cos (x)} + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\sin^{2} (x) - \cos^{2} (x)}{\cos (x) (\sin (x)) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.11.2

Batalkan faktor persekutuan.

\frac{{sin}^{2} (x) - {cos}^{2} (x)}{cos (x) sin (x) \frac{sin (x)}{cos (x)} + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\sin^{2} (x) - \cos^{2} (x)}{\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.11.3

Tulis kembali pernyataannya.

\frac{{sin}^{2} (x) - {cos}^{2} (x)}{sin (x) sin (x) + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\sin^{2} (x) - \cos^{2} (x)}{\sin (x) \sin (x) + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

\frac{{sin}^{2} (x) - {cos}^{2} (x)}{sin (x) sin (x) + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\sin^{2} (x) - \cos^{2} (x)}{\sin (x) \sin (x) + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.12

Naikkan

sin (x)

$\sin (x)$ menjadi pangkat

1

$1$ .

\frac{{sin}^{2} (x) - {cos}^{2} (x)}{{sin}^{1} (x) sin (x) + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\sin^{2} (x) - \cos^{2} (x)}{\sin^{1} (x) \sin (x) + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.13

Naikkan

sin (x)

$\sin (x)$ menjadi pangkat

1

$1$ .

\frac{{sin}^{2} (x) - {cos}^{2} (x)}{{sin}^{1} (x) {sin}^{1} (x) + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\sin^{2} (x) - \cos^{2} (x)}{\sin^{1} (x) \sin^{1} (x) + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.14

Gunakan kaidah pangkat

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ untuk menggabungkan pangkat.

\frac{{sin}^{2} (x) - {cos}^{2} (x)}{{sin (x)}^{1 + 1} + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\sin^{2} (x) - \cos^{2} (x)}{{\sin (x)}^{1 + 1} + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.15

Tambahkan

1

$1$ dan

1

$1$ .

\frac{{sin}^{2} (x) - {cos}^{2} (x)}{{sin}^{2} (x) + cos (x) sin (x) \frac{cos (x)}{sin (x)}}

$\frac{\sin^{2} (x) - \cos^{2} (x)}{\sin^{2} (x) + \cos (x) \sin (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.16

Batalkan faktor persekutuan dari

$\sin (x)$ .

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Langkah 2.5.16.1

Faktorkan

$\sin (x)$ dari

$\cos (x) \sin (x)$ .

$\frac{\sin^{2} (x) - \cos^{2} (x)}{\sin^{2} (x) + \sin (x) \cos (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.16.2

Batalkan faktor persekutuan.

$\frac{\sin^{2} (x) - \cos^{2} (x)}{\sin^{2} (x) + \sin (x) \cos (x) \frac{\cos (x)}{\sin (x)}}$

Langkah 2.5.16.3

Tulis kembali pernyataannya.

$\frac{\sin^{2} (x) - \cos^{2} (x)}{\sin^{2} (x) + \cos (x) \cos (x)}$

Langkah 2.5.17

Naikkan

$\cos (x)$ menjadi pangkat

$1$ .

$\frac{\sin^{2} (x) - \cos^{2} (x)}{\sin^{2} (x) + \cos^{1} (x) \cos (x)}$

Langkah 2.5.18

Naikkan

$\cos (x)$ menjadi pangkat

$1$ .

$\frac{\sin^{2} (x) - \cos^{2} (x)}{\sin^{2} (x) + \cos^{1} (x) \cos^{1} (x)}$

Langkah 2.5.19

Gunakan kaidah pangkat

$a^{m} a^{n} = a^{m + n}$ untuk menggabungkan pangkat.

$\frac{\sin^{2} (x) - \cos^{2} (x)}{\sin^{2} (x) + {\cos (x)}^{1 + 1}}$

Langkah 2.5.20

Tambahkan

$1$ dan

$1$ .

$\frac{\sin^{2} (x) - \cos^{2} (x)}{\sin^{2} (x) + \cos^{2} (x)}$

Langkah 2.6

Terapkan identitas pythagoras.

$\frac{\sin^{2} (x) - \cos^{2} (x)}{1}$

Langkah 2.7

Bagilah

$\sin^{2} (x) - \cos^{2} (x)$ dengan

$1$ .

$\sin^{2} (x) - \cos^{2} (x)$

Langkah 3

Terapkan identitas Pythagoras secara terbalik.

$1 - \cos^{2} (x) - \cos^{2} (x)$

Langkah 4

Kurangi

${\cos (x)}^{2}$ dengan

$- {\cos (x)}^{2}$ .

$1 - 2 \cos^{2} (x)$

Langkah 5

Karena kedua sisi telah terbukti setara, maka persamaan tersebut adalah sebuah identitas.

$\frac{\tan (x) - \cot (x)}{\tan (x) + \cot (x)} = 1 - 2 \cos^{2} (x)$ adalah identitas