حساب المثلثات الأمثلة

$\frac{1 - \sin (x)}{1 + \sin (x)} = {(\sec (x) - \tan (x))}^{2}$

خطوة 1

اطرح ${(\sec (x) - \tan (x))}^{2}$ من كلا المتعادلين.

$\frac{1 - \sin (x)}{1 + \sin (x)} - {(\sec (x) - \tan (x))}^{2} = 0$

خطوة 2

بسّط $\frac{1 - \sin (x)}{1 + \sin (x)} - {(\sec (x) - \tan (x))}^{2}$ .

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خطوة 2.1

بسّط كل حد.

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خطوة 2.1.1

بسّط كل حد.

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خطوة 2.1.1.1

أعِد كتابة $\sec (x)$ من حيث الجيوب وجيوب التمام.

$\frac{1 - \sin (x)}{1 + \sin (x)} - {(\frac{1}{\cos (x)} - \tan (x))}^{2} = 0$

خطوة 2.1.1.2

أعِد كتابة $\tan (x)$ من حيث الجيوب وجيوب التمام.

$\frac{1 - \sin (x)}{1 + \sin (x)} - {(\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)})}^{2} = 0$

خطوة 2.1.2

أعِد كتابة ${(\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)})}^{2}$ بالصيغة $(\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)}) (\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)})$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - ((\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)}) (\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)})) = 0$

خطوة 2.1.3

وسّع $(\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)}) (\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)})$ باستخدام طريقة "الأول، الخارجي، الداخلي، الأخير".

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خطوة 2.1.3.1

طبّق خاصية التوزيع.

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos (x)} (\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)}) - \frac{\sin (x)}{\cos (x)} (\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)})) = 0$

خطوة 2.1.3.2

طبّق خاصية التوزيع.

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)} + \frac{1}{\cos (x)} (- \frac{\sin (x)}{\cos (x)}) - \frac{\sin (x)}{\cos (x)} (\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)})) = 0$

خطوة 2.1.3.3

طبّق خاصية التوزيع.

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)} + \frac{1}{\cos (x)} (- \frac{\sin (x)}{\cos (x)}) - \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})) = 0$

خطوة 2.1.4

بسّط ووحّد الحدود المتشابهة.

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خطوة 2.1.4.1

بسّط كل حد.

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خطوة 2.1.4.1.1

اضرب $\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)}$ .

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خطوة 2.1.4.1.1.1

اضرب $\frac{1}{\cos (x)}$ في $\frac{1}{\cos (x)}$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos (x) \cos (x)} + \frac{1}{\cos (x)} (- \frac{\sin (x)}{\cos (x)}) - \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})) = 0$

خطوة 2.1.4.1.1.2

ارفع $\cos (x)$ إلى القوة $1$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{1} (x) \cos (x)} + \frac{1}{\cos (x)} (- \frac{\sin (x)}{\cos (x)}) - \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})) = 0$

خطوة 2.1.4.1.1.3

ارفع $\cos (x)$ إلى القوة $1$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{1} (x) \cos^{1} (x)} + \frac{1}{\cos (x)} (- \frac{\sin (x)}{\cos (x)}) - \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})) = 0$

خطوة 2.1.4.1.1.4

استخدِم قاعدة القوة $a^{m} a^{n} = a^{m + n}$ لتجميع الأُسس.

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{{\cos (x)}^{1 + 1}} + \frac{1}{\cos (x)} (- \frac{\sin (x)}{\cos (x)}) - \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})) = 0$

خطوة 2.1.4.1.1.5

أضف $1$ و $1$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} + \frac{1}{\cos (x)} (- \frac{\sin (x)}{\cos (x)}) - \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})) = 0$

خطوة 2.1.4.1.2

أعِد الكتابة باستخدام خاصية الإبدال لعملية الضرب.

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} - \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})) = 0$

خطوة 2.1.4.1.3

اضرب $- \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ .

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خطوة 2.1.4.1.3.1

اضرب $\frac{\sin (x)}{\cos (x)}$ في $\frac{1}{\cos (x)}$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{\sin (x)}{\cos (x) \cos (x)} - \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})) = 0$

خطوة 2.1.4.1.3.2

ارفع $\cos (x)$ إلى القوة $1$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{1} (x) \cos (x)} - \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})) = 0$

خطوة 2.1.4.1.3.3

ارفع $\cos (x)$ إلى القوة $1$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{1} (x) \cos^{1} (x)} - \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})) = 0$

خطوة 2.1.4.1.3.4

استخدِم قاعدة القوة $a^{m} a^{n} = a^{m + n}$ لتجميع الأُسس.

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{\sin (x)}{{\cos (x)}^{1 + 1}} - \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})) = 0$

خطوة 2.1.4.1.3.5

أضف $1$ و $1$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})) = 0$

خطوة 2.1.4.1.4

اضرب $- \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)}$ .

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خطوة 2.1.4.1.4.1

اضرب $\frac{1}{\cos (x)}$ في $\frac{\sin (x)}{\cos (x)}$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos (x) \cos (x)} - \frac{\sin (x)}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})) = 0$

خطوة 2.1.4.1.4.2

ارفع $\cos (x)$ إلى القوة $1$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{1} (x) \cos (x)} - \frac{\sin (x)}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})) = 0$

خطوة 2.1.4.1.4.3

ارفع $\cos (x)$ إلى القوة $1$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{1} (x) \cos^{1} (x)} - \frac{\sin (x)}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})) = 0$

خطوة 2.1.4.1.4.4

استخدِم قاعدة القوة $a^{m} a^{n} = a^{m + n}$ لتجميع الأُسس.

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} - \frac{\sin (x)}{{\cos (x)}^{1 + 1}} - \frac{\sin (x)}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})) = 0$

خطوة 2.1.4.1.4.5

أضف $1$ و $1$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})) = 0$

خطوة 2.1.4.1.5

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

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خطوة 2.1.4.1.5.1

اضرب $- 1$ في $- 1$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} + 1 \frac{\sin (x)}{\cos (x)} \frac{\sin (x)}{\cos (x)}) = 0$

خطوة 2.1.4.1.5.2

اضرب $\frac{\sin (x)}{\cos (x)}$ في $1$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}) = 0$

خطوة 2.1.4.1.5.3

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

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin (x) \sin (x)}{\cos (x) \cos (x)}) = 0$

خطوة 2.1.4.1.5.4

ارفع $\sin (x)$ إلى القوة $1$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{1} (x) \sin (x)}{\cos (x) \cos (x)}) = 0$

خطوة 2.1.4.1.5.5

ارفع $\sin (x)$ إلى القوة $1$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{1} (x) \sin^{1} (x)}{\cos (x) \cos (x)}) = 0$

خطوة 2.1.4.1.5.6

استخدِم قاعدة القوة $a^{m} a^{n} = a^{m + n}$ لتجميع الأُسس.

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} + \frac{{\sin (x)}^{1 + 1}}{\cos (x) \cos (x)}) = 0$

خطوة 2.1.4.1.5.7

أضف $1$ و $1$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos (x) \cos (x)}) = 0$

خطوة 2.1.4.1.5.8

ارفع $\cos (x)$ إلى القوة $1$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{1} (x) \cos (x)}) = 0$

خطوة 2.1.4.1.5.9

ارفع $\cos (x)$ إلى القوة $1$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{1} (x) \cos^{1} (x)}) = 0$

خطوة 2.1.4.1.5.10

استخدِم قاعدة القوة $a^{m} a^{n} = a^{m + n}$ لتجميع الأُسس.

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{{\cos (x)}^{1 + 1}}) = 0$

خطوة 2.1.4.1.5.11

أضف $1$ و $1$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}) = 0$

خطوة 2.1.4.2

اطرح $\frac{\sin (x)}{\cos^{2} (x)}$ من $- \frac{\sin (x)}{\cos^{2} (x)}$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - 2 \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}) = 0$

خطوة 2.1.5

بسّط كل حد.

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خطوة 2.1.5.1

اجمع $- 2$ و $\frac{\sin (x)}{\cos^{2} (x)}$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} + \frac{- 2 \sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}) = 0$

خطوة 2.1.5.2

انقُل السالب أمام الكسر.

$\frac{1 - \sin (x)}{1 + \sin (x)} - (\frac{1}{\cos^{2} (x)} - \frac{2 \sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}) = 0$

خطوة 2.1.6

طبّق خاصية التوزيع.

$\frac{1 - \sin (x)}{1 + \sin (x)} - \frac{1}{\cos^{2} (x)} - - \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.1.7

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

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خطوة 2.1.7.1

اضرب $- 1$ في $- 1$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - \frac{1}{\cos^{2} (x)} + 1 \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.1.7.2

اضرب $\frac{2 \sin (x)}{\cos^{2} (x)}$ في $1$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} - \frac{1}{\cos^{2} (x)} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.2

لكتابة $\frac{1 - \sin (x)}{1 + \sin (x)}$ على هيئة كسر بقاسم مشترك، اضرب في $\frac{\cos^{2} (x)}{\cos^{2} (x)}$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} \cdot \frac{\cos^{2} (x)}{\cos^{2} (x)} - \frac{1}{\cos^{2} (x)} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.3

لكتابة $- \frac{1}{\cos^{2} (x)}$ على هيئة كسر بقاسم مشترك، اضرب في $\frac{1 + \sin (x)}{1 + \sin (x)}$ .

$\frac{1 - \sin (x)}{1 + \sin (x)} \cdot \frac{\cos^{2} (x)}{\cos^{2} (x)} - \frac{1}{\cos^{2} (x)} \cdot \frac{1 + \sin (x)}{1 + \sin (x)} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.4

اكتب كل عبارة قاسمها المشترك $(1 + \sin (x)) \cos^{2} (x)$ ، بضربها في العامل المناسب للعدد $1$ .

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خطوة 2.4.1

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

$\frac{(1 - \sin (x)) \cos^{2} (x)}{(1 + \sin (x)) \cos^{2} (x)} - \frac{1}{\cos^{2} (x)} \cdot \frac{1 + \sin (x)}{1 + \sin (x)} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.4.2

اضرب $\frac{1}{\cos^{2} (x)}$ في $\frac{1 + \sin (x)}{1 + \sin (x)}$ .

$\frac{(1 - \sin (x)) \cos^{2} (x)}{(1 + \sin (x)) \cos^{2} (x)} - \frac{1 + \sin (x)}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.4.3

أعِد ترتيب عوامل $(1 + \sin (x)) \cos^{2} (x)$ .

$\frac{(1 - \sin (x)) \cos^{2} (x)}{\cos^{2} (x) (1 + \sin (x))} - \frac{1 + \sin (x)}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.5

اجمع البسوط على القاسم المشترك.

$\frac{(1 - \sin (x)) \cos^{2} (x) - (1 + \sin (x))}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.6

بسّط كل حد.

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خطوة 2.6.1

بسّط بَسْط الكسر.

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خطوة 2.6.1.1

طبّق خاصية التوزيع.

$\frac{1 \cos^{2} (x) - \sin (x) \cos^{2} (x) - (1 + \sin (x))}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.6.1.2

اضرب $\cos^{2} (x)$ في $1$ .

$\frac{\cos^{2} (x) - \sin (x) \cos^{2} (x) - (1 + \sin (x))}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.6.1.3

طبّق خاصية التوزيع.

$\frac{\cos^{2} (x) - \sin (x) \cos^{2} (x) - 1 \cdot 1 - \sin (x)}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.6.1.4

اضرب $- 1$ في $1$ .

$\frac{\cos^{2} (x) - \sin (x) \cos^{2} (x) - 1 - \sin (x)}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.6.1.5

أعِد كتابة $\cos^{2} (x) - \sin (x) \cos^{2} (x) - 1 - \sin (x)$ بصيغة محلّلة إلى عوامل.

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خطوة 2.6.1.5.1

أعِد تجميع الحدود.

$\frac{- \sin (x) + \cos^{2} (x) - \sin (x) \cos^{2} (x) - 1}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.6.1.5.2

أضف الأقواس.

$\frac{- \sin (x) + \cos^{2} (x) - (\sin (x) \cos^{2} (x)) - 1}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.6.1.5.3

لنفترض أن $u = \sin (x) \cos^{2} (x)$ . استبدِل $u$ بجميع حالات حدوث $\sin (x) \cos^{2} (x)$ .

$\frac{- \sin (x) - \sin^{2} (x) - u}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.6.1.5.4

أخرِج العامل $- 1$ من $- \sin^{2} (x) - u$ .

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خطوة 2.6.1.5.4.1

أعِد ترتيب $- \sin^{2} (x)$ و $- u$ .

$\frac{- \sin (x) - u - \sin^{2} (x)}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.6.1.5.4.2

أخرِج العامل $- 1$ من $- u$ .

$\frac{- \sin (x) - (u) - \sin^{2} (x)}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.6.1.5.4.3

أخرِج العامل $- 1$ من $- \sin^{2} (x)$ .

$\frac{- \sin (x) - (u) - (\sin^{2} (x))}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.6.1.5.4.4

أخرِج العامل $- 1$ من $- (u) - (\sin^{2} (x))$ .

$\frac{- \sin (x) - (u + \sin^{2} (x))}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.6.1.5.5

استبدِل كافة حالات حدوث $u$ بـ $\sin (x) \cos^{2} (x)$ .

$\frac{- \sin (x) - (\sin (x) \cos^{2} (x) + \sin^{2} (x))}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.6.1.5.6

أخرِج العامل $\sin (x)$ من $\sin (x) \cos^{2} (x) + \sin^{2} (x)$ .

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خطوة 2.6.1.5.6.1

أخرِج العامل $\sin (x)$ من $\sin (x) \cos^{2} (x)$ .

$\frac{- \sin (x) - (\sin (x) (\cos^{2} (x)) + \sin^{2} (x))}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.6.1.5.6.2

أخرِج العامل $\sin (x)$ من $\sin^{2} (x)$ .

$\frac{- \sin (x) - (\sin (x) (\cos^{2} (x)) + \sin (x) \sin (x))}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.6.1.5.6.3

أخرِج العامل $\sin (x)$ من $\sin (x) (\cos^{2} (x)) + \sin (x) \sin (x)$ .

$\frac{- \sin (x) - (\sin (x) (\cos^{2} (x) + \sin (x)))}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

$\frac{- \sin (x) - \sin (x) (\cos^{2} (x) + \sin (x))}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.6.1.5.7

أخرِج العامل $- \sin (x)$ من $- \sin (x) - \sin (x) (\cos^{2} (x) + \sin (x))$ .

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خطوة 2.6.1.5.7.1

أخرِج العامل $- \sin (x)$ من $- \sin (x)$ .

$\frac{- \sin (x) (1) - \sin (x) (\cos^{2} (x) + \sin (x))}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.6.1.5.7.2

أخرِج العامل $- \sin (x)$ من $- \sin (x) (1) - \sin (x) (\cos^{2} (x) + \sin (x))$ .

$\frac{- \sin (x) (1 + \cos^{2} (x) + \sin (x))}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.6.2

انقُل السالب أمام الكسر.

$- \frac{\sin (x) (1 + \cos^{2} (x) + \sin (x))}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.7

لكتابة $\frac{2 \sin (x)}{\cos^{2} (x)}$ على هيئة كسر بقاسم مشترك، اضرب في $\frac{1 + \sin (x)}{1 + \sin (x)}$ .

$- \frac{\sin (x) (1 + \cos^{2} (x) + \sin (x))}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x)}{\cos^{2} (x)} \cdot \frac{1 + \sin (x)}{1 + \sin (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.8

اضرب $\frac{2 \sin (x)}{\cos^{2} (x)}$ في $\frac{1 + \sin (x)}{1 + \sin (x)}$ .

$- \frac{\sin (x) (1 + \cos^{2} (x) + \sin (x))}{\cos^{2} (x) (1 + \sin (x))} + \frac{2 \sin (x) (1 + \sin (x))}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.9

اجمع البسوط على القاسم المشترك.

$\frac{- \sin (x) (1 + \cos^{2} (x) + \sin (x)) + 2 \sin (x) (1 + \sin (x))}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.10

بسّط كل حد.

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خطوة 2.10.1

بسّط بَسْط الكسر.

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خطوة 2.10.1.1

أخرِج العامل $\sin (x)$ من $- \sin (x) (1 + \cos^{2} (x) + \sin (x)) + 2 \sin (x) (1 + \sin (x))$ .

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خطوة 2.10.1.1.1

أخرِج العامل $\sin (x)$ من $- \sin (x) (1 + \cos^{2} (x) + \sin (x))$ .

$\frac{\sin (x) (- 1 (1 + \cos^{2} (x) + \sin (x))) + 2 \sin (x) (1 + \sin (x))}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.10.1.1.2

أخرِج العامل $\sin (x)$ من $2 \sin (x) (1 + \sin (x))$ .

$\frac{\sin (x) (- 1 (1 + \cos^{2} (x) + \sin (x))) + \sin (x) (2 (1 + \sin (x)))}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.10.1.1.3

أخرِج العامل $\sin (x)$ من $\sin (x) (- 1 (1 + \cos^{2} (x) + \sin (x))) + \sin (x) (2 (1 + \sin (x)))$ .

$\frac{\sin (x) (- 1 (1 + \cos^{2} (x) + \sin (x)) + 2 (1 + \sin (x)))}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.10.1.2

طبّق خاصية التوزيع.

$\frac{\sin (x) (- 1 \cdot 1 - 1 \cos^{2} (x) - 1 \sin (x) + 2 (1 + \sin (x)))}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.10.1.3

بسّط.

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خطوة 2.10.1.3.1

اضرب $- 1$ في $1$ .

$\frac{\sin (x) (- 1 - 1 \cos^{2} (x) - 1 \sin (x) + 2 (1 + \sin (x)))}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.10.1.3.2

أعِد كتابة $- 1 \cos^{2} (x)$ بالصيغة $- \cos^{2} (x)$ .

$\frac{\sin (x) (- 1 - \cos^{2} (x) - 1 \sin (x) + 2 (1 + \sin (x)))}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.10.1.3.3

أعِد كتابة $- 1 \sin (x)$ بالصيغة $- \sin (x)$ .

$\frac{\sin (x) (- 1 - \cos^{2} (x) - \sin (x) + 2 (1 + \sin (x)))}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.10.1.4

طبّق خاصية التوزيع.

$\frac{\sin (x) (- 1 - \cos^{2} (x) - \sin (x) + 2 \cdot 1 + 2 \sin (x))}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.10.1.5

اضرب $2$ في $1$ .

$\frac{\sin (x) (- 1 - \cos^{2} (x) - \sin (x) + 2 + 2 \sin (x))}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.10.1.6

أضف $- 1$ و $2$ .

$\frac{\sin (x) (1 - \cos^{2} (x) - \sin (x) + 2 \sin (x))}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.10.1.7

أضف $- \sin (x)$ و $2 \sin (x)$ .

$\frac{\sin (x) (1 - \cos^{2} (x) + \sin (x))}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.10.1.8

طبّق متطابقة فيثاغورس.

$\frac{\sin (x) (\sin^{2} (x) + \sin (x))}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.10.1.9

أخرِج العامل $\sin (x)$ من $\sin^{2} (x) + \sin (x)$ .

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خطوة 2.10.1.9.1

أخرِج العامل $\sin (x)$ من $\sin^{2} (x)$ .

$\frac{\sin (x) (\sin (x) \sin (x) + \sin (x))}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.10.1.9.2

اضرب في $1$ .

$\frac{\sin (x) (\sin (x) \sin (x) + \sin (x) \cdot 1)}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.10.1.9.3

أخرِج العامل $\sin (x)$ من $\sin (x) \sin (x) + \sin (x) \cdot 1$ .

$\frac{\sin (x) (\sin (x) (\sin (x) + 1))}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

$\frac{\sin (x) \sin (x) (\sin (x) + 1)}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.10.1.10

اجمع الأُسس.

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خطوة 2.10.1.10.1

ارفع $\sin (x)$ إلى القوة $1$ .

$\frac{\sin^{1} (x) \sin (x) (\sin (x) + 1)}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.10.1.10.2

ارفع $\sin (x)$ إلى القوة $1$ .

$\frac{\sin^{1} (x) \sin^{1} (x) (\sin (x) + 1)}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.10.1.10.3

استخدِم قاعدة القوة $a^{m} a^{n} = a^{m + n}$ لتجميع الأُسس.

$\frac{{\sin (x)}^{1 + 1} (\sin (x) + 1)}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.10.1.10.4

أضف $1$ و $1$ .

$\frac{\sin^{2} (x) (\sin (x) + 1)}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.10.2

احذِف العامل المشترك لـ $\sin (x) + 1$ و $1 + \sin (x)$ .

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خطوة 2.10.2.1

أعِد ترتيب الحدود.

$\frac{\sin^{2} (x) (1 + \sin (x))}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.10.2.2

ألغِ العامل المشترك.

$\frac{\sin^{2} (x) (1 + \sin (x))}{\cos^{2} (x) (1 + \sin (x))} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.10.2.3

أعِد كتابة العبارة.

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

خطوة 2.11

اطرح $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ من $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ .

$0 = 0$

خطوة 3

نطاق العبارة هو جميع الأعداد الحقيقية ما عدا ما يجعل العبارة غير معرّفة. في هذه الحالة، لا يوجد عدد حقيقي يجعل العبارة غير معرّفة.