حساب المثلثات الأمثلة

{(- 4 cos (x) sin (x) + 2 cos (2 x))}^{2} + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

${(- 4 \cos (x) \sin (x) + 2 \cos (2 x))}^{2} + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1

بسّط كل حد.

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خطوة 1.1

أعِد كتابة

{(- 4 cos (x) sin (x) + 2 cos (2 x))}^{2}

${(- 4 \cos (x) \sin (x) + 2 \cos (2 x))}^{2}$ بالصيغة

(- 4 cos (x) sin (x) + 2 cos (2 x)) (- 4 cos (x) sin (x) + 2 cos (2 x))

$(- 4 \cos (x) \sin (x) + 2 \cos (2 x)) (- 4 \cos (x) \sin (x) + 2 \cos (2 x))$ .

(- 4 cos (x) sin (x) + 2 cos (2 x)) (- 4 cos (x) sin (x) + 2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$(- 4 \cos (x) \sin (x) + 2 \cos (2 x)) (- 4 \cos (x) \sin (x) + 2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.2

وسّع

(- 4 cos (x) sin (x) + 2 cos (2 x)) (- 4 cos (x) sin (x) + 2 cos (2 x))

$(- 4 \cos (x) \sin (x) + 2 \cos (2 x)) (- 4 \cos (x) \sin (x) + 2 \cos (2 x))$ باستخدام طريقة "الأول، الخارجي، الداخلي، الأخير".

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خطوة 1.2.1

طبّق خاصية التوزيع.

- 4 cos (x) sin (x) (- 4 cos (x) sin (x) + 2 cos (2 x)) + 2 cos (2 x) (- 4 cos (x) sin (x) + 2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$- 4 \cos (x) \sin (x) (- 4 \cos (x) \sin (x) + 2 \cos (2 x)) + 2 \cos (2 x) (- 4 \cos (x) \sin (x) + 2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.2.2

طبّق خاصية التوزيع.

- 4 cos (x) sin (x) (- 4 cos (x) sin (x)) - 4 cos (x) sin (x) (2 cos (2 x)) + 2 cos (2 x) (- 4 cos (x) sin (x) + 2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$- 4 \cos (x) \sin (x) (- 4 \cos (x) \sin (x)) - 4 \cos (x) \sin (x) (2 \cos (2 x)) + 2 \cos (2 x) (- 4 \cos (x) \sin (x) + 2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.2.3

طبّق خاصية التوزيع.

- 4 cos (x) sin (x) (- 4 cos (x) sin (x)) - 4 cos (x) sin (x) (2 cos (2 x)) + 2 cos (2 x) (- 4 cos (x) sin (x)) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$- 4 \cos (x) \sin (x) (- 4 \cos (x) \sin (x)) - 4 \cos (x) \sin (x) (2 \cos (2 x)) + 2 \cos (2 x) (- 4 \cos (x) \sin (x)) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

- 4 cos (x) sin (x) (- 4 cos (x) sin (x)) - 4 cos (x) sin (x) (2 cos (2 x)) + 2 cos (2 x) (- 4 cos (x) sin (x)) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$- 4 \cos (x) \sin (x) (- 4 \cos (x) \sin (x)) - 4 \cos (x) \sin (x) (2 \cos (2 x)) + 2 \cos (2 x) (- 4 \cos (x) \sin (x)) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3

بسّط ووحّد الحدود المتشابهة.

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خطوة 1.3.1

بسّط كل حد.

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خطوة 1.3.1.1

اضرب

- 4 cos (x) sin (x) (- 4 cos (x) sin (x))

$- 4 \cos (x) \sin (x) (- 4 \cos (x) \sin (x))$ .

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خطوة 1.3.1.1.1

اضرب

- 4

$- 4$ في

- 4

$- 4$ .

16 cos (x) sin (x) (cos (x) sin (x)) - 4 cos (x) sin (x) (2 cos (2 x)) + 2 cos (2 x) (- 4 cos (x) sin (x)) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos (x) \sin (x) (\cos (x) \sin (x)) - 4 \cos (x) \sin (x) (2 \cos (2 x)) + 2 \cos (2 x) (- 4 \cos (x) \sin (x)) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.1.2

ارفع

cos (x)

$\cos (x)$ إلى القوة

1

$1$ .

16 ({cos}^{1} (x) cos (x)) sin (x) sin (x) - 4 cos (x) sin (x) (2 cos (2 x)) + 2 cos (2 x) (- 4 cos (x) sin (x)) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 (\cos^{1} (x) \cos (x)) \sin (x) \sin (x) - 4 \cos (x) \sin (x) (2 \cos (2 x)) + 2 \cos (2 x) (- 4 \cos (x) \sin (x)) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.1.3

ارفع

cos (x)

$\cos (x)$ إلى القوة

1

$1$ .

16 ({cos}^{1} (x) {cos}^{1} (x)) sin (x) sin (x) - 4 cos (x) sin (x) (2 cos (2 x)) + 2 cos (2 x) (- 4 cos (x) sin (x)) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 (\cos^{1} (x) \cos^{1} (x)) \sin (x) \sin (x) - 4 \cos (x) \sin (x) (2 \cos (2 x)) + 2 \cos (2 x) (- 4 \cos (x) \sin (x)) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.1.4

استخدِم قاعدة القوة

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

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

16 {cos (x)}^{1 + 1} sin (x) sin (x) - 4 cos (x) sin (x) (2 cos (2 x)) + 2 cos (2 x) (- 4 cos (x) sin (x)) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 {\cos (x)}^{1 + 1} \sin (x) \sin (x) - 4 \cos (x) \sin (x) (2 \cos (2 x)) + 2 \cos (2 x) (- 4 \cos (x) \sin (x)) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.1.5

أضف

1

$1$ و

1

$1$ .

16 {cos}^{2} (x) sin (x) sin (x) - 4 cos (x) sin (x) (2 cos (2 x)) + 2 cos (2 x) (- 4 cos (x) sin (x)) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin (x) \sin (x) - 4 \cos (x) \sin (x) (2 \cos (2 x)) + 2 \cos (2 x) (- 4 \cos (x) \sin (x)) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.1.6

ارفع

sin (x)

$\sin (x)$ إلى القوة

1

$1$ .

16 {cos}^{2} (x) ({sin}^{1} (x) sin (x)) - 4 cos (x) sin (x) (2 cos (2 x)) + 2 cos (2 x) (- 4 cos (x) sin (x)) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) (\sin^{1} (x) \sin (x)) - 4 \cos (x) \sin (x) (2 \cos (2 x)) + 2 \cos (2 x) (- 4 \cos (x) \sin (x)) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.1.7

ارفع

sin (x)

$\sin (x)$ إلى القوة

1

$1$ .

16 {cos}^{2} (x) ({sin}^{1} (x) {sin}^{1} (x)) - 4 cos (x) sin (x) (2 cos (2 x)) + 2 cos (2 x) (- 4 cos (x) sin (x)) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) (\sin^{1} (x) \sin^{1} (x)) - 4 \cos (x) \sin (x) (2 \cos (2 x)) + 2 \cos (2 x) (- 4 \cos (x) \sin (x)) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.1.8

استخدِم قاعدة القوة

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

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

16 {cos}^{2} (x) {sin (x)}^{1 + 1} - 4 cos (x) sin (x) (2 cos (2 x)) + 2 cos (2 x) (- 4 cos (x) sin (x)) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) {\sin (x)}^{1 + 1} - 4 \cos (x) \sin (x) (2 \cos (2 x)) + 2 \cos (2 x) (- 4 \cos (x) \sin (x)) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.1.9

أضف

1

$1$ و

1

$1$ .

16 {cos}^{2} (x) {sin}^{2} (x) - 4 cos (x) sin (x) (2 cos (2 x)) + 2 cos (2 x) (- 4 cos (x) sin (x)) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin^{2} (x) - 4 \cos (x) \sin (x) (2 \cos (2 x)) + 2 \cos (2 x) (- 4 \cos (x) \sin (x)) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

16 {cos}^{2} (x) {sin}^{2} (x) - 4 cos (x) sin (x) (2 cos (2 x)) + 2 cos (2 x) (- 4 cos (x) sin (x)) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin^{2} (x) - 4 \cos (x) \sin (x) (2 \cos (2 x)) + 2 \cos (2 x) (- 4 \cos (x) \sin (x)) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.2

أضف الأقواس.

16 {cos}^{2} (x) {sin}^{2} (x) - 4 cos (x) (sin (x) \cdot 2) cos (2 x) + 2 cos (2 x) (- 4 cos (x) sin (x)) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin^{2} (x) - 4 \cos (x) (\sin (x) \cdot 2) \cos (2 x) + 2 \cos (2 x) (- 4 \cos (x) \sin (x)) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.3

أضف الأقواس.

16 {cos}^{2} (x) {sin}^{2} (x) - 4 (cos (x) (sin (x) \cdot 2)) cos (2 x) + 2 cos (2 x) (- 4 cos (x) sin (x)) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin^{2} (x) - 4 (\cos (x) (\sin (x) \cdot 2)) \cos (2 x) + 2 \cos (2 x) (- 4 \cos (x) \sin (x)) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.4

أعِد ترتيب

cos (x)

$\cos (x)$ و

sin (x) \cdot 2

$\sin (x) \cdot 2$ .

16 {cos}^{2} (x) {sin}^{2} (x) - 4 (sin (x) \cdot 2 cos (x)) cos (2 x) + 2 cos (2 x) (- 4 cos (x) sin (x)) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin^{2} (x) - 4 (\sin (x) \cdot 2 \cos (x)) \cos (2 x) + 2 \cos (2 x) (- 4 \cos (x) \sin (x)) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.5

أعِد ترتيب

sin (x)

$\sin (x)$ و

2

$2$ .

16 {cos}^{2} (x) {sin}^{2} (x) - 4 (2 \cdot sin (x) cos (x)) cos (2 x) + 2 cos (2 x) (- 4 cos (x) sin (x)) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin^{2} (x) - 4 (2 \cdot \sin (x) \cos (x)) \cos (2 x) + 2 \cos (2 x) (- 4 \cos (x) \sin (x)) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.6

طبّق متطابقة ضعف الزاوية للجيب.

16 {cos}^{2} (x) {sin}^{2} (x) - 4 sin (2 x) cos (2 x) + 2 cos (2 x) (- 4 cos (x) sin (x)) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin^{2} (x) - 4 \sin (2 x) \cos (2 x) + 2 \cos (2 x) (- 4 \cos (x) \sin (x)) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.7

أعِد ترتيب

2 cos (2 x)

$2 \cos (2 x)$ و

- 4 cos (x) sin (x)

$- 4 \cos (x) \sin (x)$ .

16 {cos}^{2} (x) {sin}^{2} (x) - 4 sin (2 x) cos (2 x) - 4 cos (x) sin (x) (2 cos (2 x)) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin^{2} (x) - 4 \sin (2 x) \cos (2 x) - 4 \cos (x) \sin (x) (2 \cos (2 x)) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.8

أضف الأقواس.

16 {cos}^{2} (x) {sin}^{2} (x) - 4 sin (2 x) cos (2 x) - 4 cos (x) (sin (x) \cdot 2) cos (2 x) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin^{2} (x) - 4 \sin (2 x) \cos (2 x) - 4 \cos (x) (\sin (x) \cdot 2) \cos (2 x) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.9

أضف الأقواس.

16 {cos}^{2} (x) {sin}^{2} (x) - 4 sin (2 x) cos (2 x) - 4 (cos (x) (sin (x) \cdot 2)) cos (2 x) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin^{2} (x) - 4 \sin (2 x) \cos (2 x) - 4 (\cos (x) (\sin (x) \cdot 2)) \cos (2 x) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.10

أعِد ترتيب

cos (x)

$\cos (x)$ و

sin (x) \cdot 2

$\sin (x) \cdot 2$ .

16 {cos}^{2} (x) {sin}^{2} (x) - 4 sin (2 x) cos (2 x) - 4 (sin (x) \cdot 2 cos (x)) cos (2 x) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin^{2} (x) - 4 \sin (2 x) \cos (2 x) - 4 (\sin (x) \cdot 2 \cos (x)) \cos (2 x) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.11

أعِد ترتيب

sin (x)

$\sin (x)$ و

2

$2$ .

16 {cos}^{2} (x) {sin}^{2} (x) - 4 sin (2 x) cos (2 x) - 4 (2 \cdot sin (x) cos (x)) cos (2 x) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin^{2} (x) - 4 \sin (2 x) \cos (2 x) - 4 (2 \cdot \sin (x) \cos (x)) \cos (2 x) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.12

طبّق متطابقة ضعف الزاوية للجيب.

16 {cos}^{2} (x) {sin}^{2} (x) - 4 sin (2 x) cos (2 x) - 4 sin (2 x) cos (2 x) + 2 cos (2 x) (2 cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin^{2} (x) - 4 \sin (2 x) \cos (2 x) - 4 \sin (2 x) \cos (2 x) + 2 \cos (2 x) (2 \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.13

اضرب

2 cos (2 x) (2 cos (2 x))

$2 \cos (2 x) (2 \cos (2 x))$ .

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خطوة 1.3.1.13.1

اضرب

2

$2$ في

2

$2$ .

16 {cos}^{2} (x) {sin}^{2} (x) - 4 sin (2 x) cos (2 x) - 4 sin (2 x) cos (2 x) + 4 cos (2 x) cos (2 x) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin^{2} (x) - 4 \sin (2 x) \cos (2 x) - 4 \sin (2 x) \cos (2 x) + 4 \cos (2 x) \cos (2 x) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.13.2

ارفع

cos (2 x)

$\cos (2 x)$ إلى القوة

1

$1$ .

16 {cos}^{2} (x) {sin}^{2} (x) - 4 sin (2 x) cos (2 x) - 4 sin (2 x) cos (2 x) + 4 ({cos}^{1} (2 x) cos (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin^{2} (x) - 4 \sin (2 x) \cos (2 x) - 4 \sin (2 x) \cos (2 x) + 4 (\cos^{1} (2 x) \cos (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.13.3

ارفع

cos (2 x)

$\cos (2 x)$ إلى القوة

1

$1$ .

16 {cos}^{2} (x) {sin}^{2} (x) - 4 sin (2 x) cos (2 x) - 4 sin (2 x) cos (2 x) + 4 ({cos}^{1} (2 x) {cos}^{1} (2 x)) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin^{2} (x) - 4 \sin (2 x) \cos (2 x) - 4 \sin (2 x) \cos (2 x) + 4 (\cos^{1} (2 x) \cos^{1} (2 x)) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.13.4

استخدِم قاعدة القوة

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

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

16 {cos}^{2} (x) {sin}^{2} (x) - 4 sin (2 x) cos (2 x) - 4 sin (2 x) cos (2 x) + 4 {cos (2 x)}^{1 + 1} + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin^{2} (x) - 4 \sin (2 x) \cos (2 x) - 4 \sin (2 x) \cos (2 x) + 4 {\cos (2 x)}^{1 + 1} + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.1.13.5

أضف

1

$1$ و

1

$1$ .

16 {cos}^{2} (x) {sin}^{2} (x) - 4 sin (2 x) cos (2 x) - 4 sin (2 x) cos (2 x) + 4 {cos}^{2} (2 x) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin^{2} (x) - 4 \sin (2 x) \cos (2 x) - 4 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

16 {cos}^{2} (x) {sin}^{2} (x) - 4 sin (2 x) cos (2 x) - 4 sin (2 x) cos (2 x) + 4 {cos}^{2} (2 x) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin^{2} (x) - 4 \sin (2 x) \cos (2 x) - 4 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

16 {cos}^{2} (x) {sin}^{2} (x) - 4 sin (2 x) cos (2 x) - 4 sin (2 x) cos (2 x) + 4 {cos}^{2} (2 x) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin^{2} (x) - 4 \sin (2 x) \cos (2 x) - 4 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.3.2

اطرح

4 sin (2 x) cos (2 x)

$4 \sin (2 x) \cos (2 x)$ من

- 4 sin (2 x) cos (2 x)

$- 4 \sin (2 x) \cos (2 x)$ .

16 {cos}^{2} (x) {sin}^{2} (x) - 8 sin (2 x) cos (2 x) + 4 {cos}^{2} (2 x) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

16 {cos}^{2} (x) {sin}^{2} (x) - 8 sin (2 x) cos (2 x) + 4 {cos}^{2} (2 x) + {(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + {(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$

خطوة 1.4

أعِد كتابة

{(2 cos (2 x) + 4 sin (x) cos (x))}^{2}

${(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$ بالصيغة

(2 cos (2 x) + 4 sin (x) cos (x)) (2 cos (2 x) + 4 sin (x) cos (x))

$(2 \cos (2 x) + 4 \sin (x) \cos (x)) (2 \cos (2 x) + 4 \sin (x) \cos (x))$ .

16 {cos}^{2} (x) {sin}^{2} (x) - 8 sin (2 x) cos (2 x) + 4 {cos}^{2} (2 x) + (2 cos (2 x) + 4 sin (x) cos (x)) (2 cos (2 x) + 4 sin (x) cos (x))

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + (2 \cos (2 x) + 4 \sin (x) \cos (x)) (2 \cos (2 x) + 4 \sin (x) \cos (x))$

خطوة 1.5

وسّع

(2 cos (2 x) + 4 sin (x) cos (x)) (2 cos (2 x) + 4 sin (x) cos (x))

$(2 \cos (2 x) + 4 \sin (x) \cos (x)) (2 \cos (2 x) + 4 \sin (x) \cos (x))$ باستخدام طريقة "الأول، الخارجي، الداخلي، الأخير".

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خطوة 1.5.1

طبّق خاصية التوزيع.

16 {cos}^{2} (x) {sin}^{2} (x) - 8 sin (2 x) cos (2 x) + 4 {cos}^{2} (2 x) + 2 cos (2 x) (2 cos (2 x) + 4 sin (x) cos (x)) + 4 sin (x) cos (x) (2 cos (2 x) + 4 sin (x) cos (x))

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 2 \cos (2 x) (2 \cos (2 x) + 4 \sin (x) \cos (x)) + 4 \sin (x) \cos (x) (2 \cos (2 x) + 4 \sin (x) \cos (x))$

خطوة 1.5.2

طبّق خاصية التوزيع.

16 {cos}^{2} (x) {sin}^{2} (x) - 8 sin (2 x) cos (2 x) + 4 {cos}^{2} (2 x) + 2 cos (2 x) (2 cos (2 x)) + 2 cos (2 x) (4 sin (x) cos (x)) + 4 sin (x) cos (x) (2 cos (2 x) + 4 sin (x) cos (x))

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 2 \cos (2 x) (2 \cos (2 x)) + 2 \cos (2 x) (4 \sin (x) \cos (x)) + 4 \sin (x) \cos (x) (2 \cos (2 x) + 4 \sin (x) \cos (x))$

خطوة 1.5.3

طبّق خاصية التوزيع.

16 {cos}^{2} (x) {sin}^{2} (x) - 8 sin (2 x) cos (2 x) + 4 {cos}^{2} (2 x) + 2 cos (2 x) (2 cos (2 x)) + 2 cos (2 x) (4 sin (x) cos (x)) + 4 sin (x) cos (x) (2 cos (2 x)) + 4 sin (x) cos (x) (4 sin (x) cos (x))

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 2 \cos (2 x) (2 \cos (2 x)) + 2 \cos (2 x) (4 \sin (x) \cos (x)) + 4 \sin (x) \cos (x) (2 \cos (2 x)) + 4 \sin (x) \cos (x) (4 \sin (x) \cos (x))$

16 {cos}^{2} (x) {sin}^{2} (x) - 8 sin (2 x) cos (2 x) + 4 {cos}^{2} (2 x) + 2 cos (2 x) (2 cos (2 x)) + 2 cos (2 x) (4 sin (x) cos (x)) + 4 sin (x) cos (x) (2 cos (2 x)) + 4 sin (x) cos (x) (4 sin (x) cos (x))

خطوة 1.6

بسّط ووحّد الحدود المتشابهة.

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خطوة 1.6.1

بسّط كل حد.

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خطوة 1.6.1.1

اضرب

2 cos (2 x) (2 cos (2 x))

$2 \cos (2 x) (2 \cos (2 x))$ .

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خطوة 1.6.1.1.1

اضرب

2

$2$ في

2

$2$ .

16 {cos}^{2} (x) {sin}^{2} (x) - 8 sin (2 x) cos (2 x) + 4 {cos}^{2} (2 x) + 4 cos (2 x) cos (2 x) + 2 cos (2 x) (4 sin (x) cos (x)) + 4 sin (x) cos (x) (2 cos (2 x)) + 4 sin (x) cos (x) (4 sin (x) cos (x))

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos (2 x) \cos (2 x) + 2 \cos (2 x) (4 \sin (x) \cos (x)) + 4 \sin (x) \cos (x) (2 \cos (2 x)) + 4 \sin (x) \cos (x) (4 \sin (x) \cos (x))$

خطوة 1.6.1.1.2

ارفع

cos (2 x)

$\cos (2 x)$ إلى القوة

1

$1$ .

16 {cos}^{2} (x) {sin}^{2} (x) - 8 sin (2 x) cos (2 x) + 4 {cos}^{2} (2 x) + 4 ({cos}^{1} (2 x) cos (2 x)) + 2 cos (2 x) (4 sin (x) cos (x)) + 4 sin (x) cos (x) (2 cos (2 x)) + 4 sin (x) cos (x) (4 sin (x) cos (x))

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 (\cos^{1} (2 x) \cos (2 x)) + 2 \cos (2 x) (4 \sin (x) \cos (x)) + 4 \sin (x) \cos (x) (2 \cos (2 x)) + 4 \sin (x) \cos (x) (4 \sin (x) \cos (x))$

خطوة 1.6.1.1.3

ارفع

cos (2 x)

$\cos (2 x)$ إلى القوة

1

$1$ .

16 {cos}^{2} (x) {sin}^{2} (x) - 8 sin (2 x) cos (2 x) + 4 {cos}^{2} (2 x) + 4 ({cos}^{1} (2 x) {cos}^{1} (2 x)) + 2 cos (2 x) (4 sin (x) cos (x)) + 4 sin (x) cos (x) (2 cos (2 x)) + 4 sin (x) cos (x) (4 sin (x) cos (x))

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 (\cos^{1} (2 x) \cos^{1} (2 x)) + 2 \cos (2 x) (4 \sin (x) \cos (x)) + 4 \sin (x) \cos (x) (2 \cos (2 x)) + 4 \sin (x) \cos (x) (4 \sin (x) \cos (x))$

خطوة 1.6.1.1.4

استخدِم قاعدة القوة

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

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

16 {cos}^{2} (x) {sin}^{2} (x) - 8 sin (2 x) cos (2 x) + 4 {cos}^{2} (2 x) + 4 {cos (2 x)}^{1 + 1} + 2 cos (2 x) (4 sin (x) cos (x)) + 4 sin (x) cos (x) (2 cos (2 x)) + 4 sin (x) cos (x) (4 sin (x) cos (x))

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 {\cos (2 x)}^{1 + 1} + 2 \cos (2 x) (4 \sin (x) \cos (x)) + 4 \sin (x) \cos (x) (2 \cos (2 x)) + 4 \sin (x) \cos (x) (4 \sin (x) \cos (x))$

خطوة 1.6.1.1.5

أضف

1

$1$ و

1

$1$ .

16 {cos}^{2} (x) {sin}^{2} (x) - 8 sin (2 x) cos (2 x) + 4 {cos}^{2} (2 x) + 4 {cos}^{2} (2 x) + 2 cos (2 x) (4 sin (x) cos (x)) + 4 sin (x) cos (x) (2 cos (2 x)) + 4 sin (x) cos (x) (4 sin (x) cos (x))

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 2 \cos (2 x) (4 \sin (x) \cos (x)) + 4 \sin (x) \cos (x) (2 \cos (2 x)) + 4 \sin (x) \cos (x) (4 \sin (x) \cos (x))$

16 {cos}^{2} (x) {sin}^{2} (x) - 8 sin (2 x) cos (2 x) + 4 {cos}^{2} (2 x) + 4 {cos}^{2} (2 x) + 2 cos (2 x) (4 sin (x) cos (x)) + 4 sin (x) cos (x) (2 cos (2 x)) + 4 sin (x) cos (x) (4 sin (x) cos (x))

خطوة 1.6.1.2

أعِد ترتيب

2 cos (2 x)

$2 \cos (2 x)$ و

4 sin (x) cos (x)

$4 \sin (x) \cos (x)$ .

16 {cos}^{2} (x) {sin}^{2} (x) - 8 sin (2 x) cos (2 x) + 4 {cos}^{2} (2 x) + 4 {cos}^{2} (2 x) + 4 sin (x) cos (x) (2 cos (2 x)) + 4 sin (x) cos (x) (2 cos (2 x)) + 4 sin (x) cos (x) (4 sin (x) cos (x))

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 4 \sin (x) \cos (x) (2 \cos (2 x)) + 4 \sin (x) \cos (x) (2 \cos (2 x)) + 4 \sin (x) \cos (x) (4 \sin (x) \cos (x))$

خطوة 1.6.1.3

أضف الأقواس.

16 {cos}^{2} (x) {sin}^{2} (x) - 8 sin (2 x) cos (2 x) + 4 {cos}^{2} (2 x) + 4 {cos}^{2} (2 x) + 4 sin (x) (cos (x) \cdot 2) cos (2 x) + 4 sin (x) cos (x) (2 cos (2 x)) + 4 sin (x) cos (x) (4 sin (x) cos (x))

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 4 \sin (x) (\cos (x) \cdot 2) \cos (2 x) + 4 \sin (x) \cos (x) (2 \cos (2 x)) + 4 \sin (x) \cos (x) (4 \sin (x) \cos (x))$

خطوة 1.6.1.4

أضف الأقواس.

16 {cos}^{2} (x) {sin}^{2} (x) - 8 sin (2 x) cos (2 x) + 4 {cos}^{2} (2 x) + 4 {cos}^{2} (2 x) + 4 (sin (x) (cos (x) \cdot 2)) cos (2 x) + 4 sin (x) cos (x) (2 cos (2 x)) + 4 sin (x) cos (x) (4 sin (x) cos (x))

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 4 (\sin (x) (\cos (x) \cdot 2)) \cos (2 x) + 4 \sin (x) \cos (x) (2 \cos (2 x)) + 4 \sin (x) \cos (x) (4 \sin (x) \cos (x))$

خطوة 1.6.1.5

أعِد ترتيب

$\sin (x) \cos (x)$ و

$2$ .

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 4 (2 \cdot (\sin (x) \cos (x))) \cos (2 x) + 4 \sin (x) \cos (x) (2 \cos (2 x)) + 4 \sin (x) \cos (x) (4 \sin (x) \cos (x))$

خطوة 1.6.1.6

طبّق متطابقة ضعف الزاوية للجيب.

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 4 \sin (2 x) \cos (2 x) + 4 \sin (x) \cos (x) (2 \cos (2 x)) + 4 \sin (x) \cos (x) (4 \sin (x) \cos (x))$

خطوة 1.6.1.7

أضف الأقواس.

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 4 \sin (2 x) \cos (2 x) + 4 \sin (x) (\cos (x) \cdot 2) \cos (2 x) + 4 \sin (x) \cos (x) (4 \sin (x) \cos (x))$

خطوة 1.6.1.8

أضف الأقواس.

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 4 \sin (2 x) \cos (2 x) + 4 (\sin (x) (\cos (x) \cdot 2)) \cos (2 x) + 4 \sin (x) \cos (x) (4 \sin (x) \cos (x))$

خطوة 1.6.1.9

أعِد ترتيب

$\sin (x) \cos (x)$ و

$2$ .

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 4 \sin (2 x) \cos (2 x) + 4 (2 \cdot (\sin (x) \cos (x))) \cos (2 x) + 4 \sin (x) \cos (x) (4 \sin (x) \cos (x))$

خطوة 1.6.1.10

طبّق متطابقة ضعف الزاوية للجيب.

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 4 \sin (2 x) \cos (2 x) + 4 \sin (2 x) \cos (2 x) + 4 \sin (x) \cos (x) (4 \sin (x) \cos (x))$

خطوة 1.6.1.11

اضرب

$4 \sin (x) \cos (x) (4 \sin (x) \cos (x))$ .

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خطوة 1.6.1.11.1

اضرب

$4$ في

$4$ .

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 4 \sin (2 x) \cos (2 x) + 4 \sin (2 x) \cos (2 x) + 16 \sin (x) \cos (x) (\sin (x) \cos (x))$

خطوة 1.6.1.11.2

ارفع

$\sin (x)$ إلى القوة

$1$ .

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 4 \sin (2 x) \cos (2 x) + 4 \sin (2 x) \cos (2 x) + 16 (\sin^{1} (x) \sin (x)) \cos (x) \cos (x)$

خطوة 1.6.1.11.3

ارفع

$\sin (x)$ إلى القوة

$1$ .

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 4 \sin (2 x) \cos (2 x) + 4 \sin (2 x) \cos (2 x) + 16 (\sin^{1} (x) \sin^{1} (x)) \cos (x) \cos (x)$

خطوة 1.6.1.11.4

استخدِم قاعدة القوة

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

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 4 \sin (2 x) \cos (2 x) + 4 \sin (2 x) \cos (2 x) + 16 {\sin (x)}^{1 + 1} \cos (x) \cos (x)$

خطوة 1.6.1.11.5

أضف

$1$ و

$1$ .

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 4 \sin (2 x) \cos (2 x) + 4 \sin (2 x) \cos (2 x) + 16 \sin^{2} (x) \cos (x) \cos (x)$

خطوة 1.6.1.11.6

ارفع

$\cos (x)$ إلى القوة

$1$ .

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 4 \sin (2 x) \cos (2 x) + 4 \sin (2 x) \cos (2 x) + 16 \sin^{2} (x) (\cos^{1} (x) \cos (x))$

خطوة 1.6.1.11.7

ارفع

$\cos (x)$ إلى القوة

$1$ .

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 4 \sin (2 x) \cos (2 x) + 4 \sin (2 x) \cos (2 x) + 16 \sin^{2} (x) (\cos^{1} (x) \cos^{1} (x))$

خطوة 1.6.1.11.8

استخدِم قاعدة القوة

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

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 4 \sin (2 x) \cos (2 x) + 4 \sin (2 x) \cos (2 x) + 16 \sin^{2} (x) {\cos (x)}^{1 + 1}$

خطوة 1.6.1.11.9

أضف

$1$ و

$1$ .

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 4 \sin (2 x) \cos (2 x) + 4 \sin (2 x) \cos (2 x) + 16 \sin^{2} (x) \cos^{2} (x)$

خطوة 1.6.2

أضف

$4 \sin (2 x) \cos (2 x)$ و

$4 \sin (2 x) \cos (2 x)$ .

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 8 \sin (2 x) \cos (2 x) + 16 \sin^{2} (x) \cos^{2} (x)$

خطوة 2

بسّط بجمع الحدود.

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خطوة 2.1

جمّع الحدود المتعاكسة في

$16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 8 \sin (2 x) \cos (2 x) + 16 \sin^{2} (x) \cos^{2} (x)$ .

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خطوة 2.1.1

أضف

$- 8 \sin (2 x) \cos (2 x)$ و

$8 \sin (2 x) \cos (2 x)$ .

$16 \cos^{2} (x) \sin^{2} (x) + 0 + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 16 \sin^{2} (x) \cos^{2} (x)$

خطوة 2.1.2

أضف

$16 \cos^{2} (x) \sin^{2} (x)$ و

$0$ .

$16 \cos^{2} (x) \sin^{2} (x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 16 \sin^{2} (x) \cos^{2} (x)$

خطوة 2.2

أعِد ترتيب عوامل

$16 \sin^{2} (x) \cos^{2} (x)$ .

$16 \cos^{2} (x) \sin^{2} (x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 16 \cos^{2} (x) \sin^{2} (x)$

خطوة 2.3

أضف

$16 \cos^{2} (x) \sin^{2} (x)$ و

$16 \cos^{2} (x) \sin^{2} (x)$ .

$32 \cos^{2} (x) \sin^{2} (x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x)$

خطوة 2.4

أضف

$4 \cos^{2} (2 x)$ و

$4 \cos^{2} (2 x)$ .

$32 \cos^{2} (x) \sin^{2} (x) + 8 \cos^{2} (2 x)$