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العربية

prove (2sin(x)+cos(x))^2+(2cos(x)-sin(x))^2=5

الحلّ

prove

(2 sin (x) + cos (x))^{2} + (2 cos (x) - sin (x))^{2} = 5

صحيح

خطوات الحلّ

(2 sin (x) + cos (x))^{2} + (2 cos (x) - sin (x))^{2} = 5

قم بالعمل على الطرف الأيسر

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

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

حلل إلى عوامل:

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

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

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

(a + b)^{2} = a^{2} + 2 ab + b^{2}

:فعّل صيغة الضرب المختصر

a = - sin (x), b = 2 cos (x)

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

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

بسّط:

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

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

(- a) = - a

:احذف الأقواس

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

(- sin (x))^{2} = sin^{2} (x)

(- sin (x))^{2}

زوجيّ n

إذا تحقّق أنّ

, (- a)^{n} = a^{n}

:فعّل قانون القوى

(- sin (x))^{2} = sin^{2} (x)

= sin^{2} (x)

2 sin (x) \cdot 2 cos (x) = 4 sin (x) cos (x)

2 sin (x) \cdot 2 cos (x)

2 \cdot 2 = 4 :

اضرب الأعداد

= 4 sin (x) cos (x)

(2 cos (x))^{2} = 4 cos^{2} (x)

(2 cos (x))^{2}

(a \cdot b)^{n} = a^{n} b^{n}

:فعّل قانون القوى

= 2^{2} cos^{2} (x)

2^{2} = 4

= 4 cos^{2} (x)

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

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

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

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

(a + b)^{2} = a^{2} + 2 ab + b^{2}

:فعّل صيغة الضرب المختصر

a = cos (x), b = 2 sin (x)

= cos^{2} (x) + 2 cos (x) \cdot 2 sin (x) + (2 sin (x))^{2}

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

بسّط:

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

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

2 cos (x) \cdot 2 sin (x) = 4 cos (x) sin (x)

2 cos (x) \cdot 2 sin (x)

2 \cdot 2 = 4 :

اضرب الأعداد

= 4 cos (x) sin (x)

(2 sin (x))^{2} = 4 sin^{2} (x)

(2 sin (x))^{2}

(a \cdot b)^{n} = a^{n} b^{n}

:فعّل قانون القوى

= 2^{2} sin^{2} (x)

2^{2} = 4

= 4 sin^{2} (x)

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

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

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

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

بسّط:

5 sin^{2} (x) + 5 cos^{2} (x)

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

- 4 sin (x) cos (x) + 4 cos (x) sin (x) = 0 :

اجمع العناصر المتشابهة

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

4 cos^{2} (x) + cos^{2} (x) = 5 cos^{2} (x) :

اجمع العناصر المتشابهة

= sin^{2} (x) + 5 cos^{2} (x) + 4 sin^{2} (x)

sin^{2} (x) + 4 sin^{2} (x) = 5 sin^{2} (x) :

اجمع العناصر المتشابهة

= 5 sin^{2} (x) + 5 cos^{2} (x)

= 5 sin^{2} (x) + 5 cos^{2} (x)

5

قم باخراج العامل المشترك

= 5 (sin^{2} (x) + cos^{2} (x))

= (cos^{2} (x) + sin^{2} (x)) \cdot 5

Rewrite using trig identities

(cos^{2} (x) + sin^{2} (x)) \cdot 5

cos^{2} (x) + sin^{2} (x) = 1

:فعّل نطريّة فيتاغوروس

= 1 \cdot 5

بسّط

= 5

= 5

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e cos (a) (sec (a) - cos (a)) = sin^{2} (a)

p ro v e tan (x) cos (x) csc (x) = 1

p ro v e \frac{csc ^{4} ( x ) - 1}{1 + csc ^{2} ( x )} = cot^{2} (x)

p ro v e 1 + sin (2 θ) = (cos (θ) sin (θ))^{2}

p ro v e cos^{2} (x) = \frac{1}{2} (cos (2 x) + 1)