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العربية

prove sin^4(x)-2sin^2(x)=cos^4(x)-1

الحلّ

prove

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

صحيح

خطوات الحلّ

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

قم بالعمل على الطرف الأيسر

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

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

حلل إلى عوامل:

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

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

a^{b + c} = a^{b} a^{c}

:فعّل قانون القوى

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

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

sin^{2} (x)

قم باخراج العامل المشترك

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

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

Rewrite using trig identities

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

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

:فعّل نطريّة فيتاغوروس

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

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

بسّط

= (- cos^{2} (x) - 1) (1 - cos^{2} (x))

= (- cos^{2} (x) - 1) (1 - cos^{2} (x))

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

وسٌع:

- 1 + cos^{4} (x)

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

(a + b) (c + d) = a c + a d + b c + b d

(a + b) (c + d) = a c + a d + b c + b d

فعّل قانون التوزيع

a = - 1, b = - cos^{2} (x), c = 1, d = - cos^{2} (x)

= (- 1) \cdot 1 + (- 1) (- cos^{2} (x)) + (- cos^{2} (x)) \cdot 1 + (- cos^{2} (x)) (- cos^{2} (x))

فعّل قوانين سالب-موجب

+ (- a) = - a, (- a) (- b) = ab

= - 1 \cdot 1 + 1 \cdot cos^{2} (x) - 1 \cdot cos^{2} (x) + cos^{2} (x) cos^{2} (x)

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

بسّط:

- 1 + cos^{4} (x)

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

1 \cdot cos^{2} (x) - 1 \cdot cos^{2} (x) = 0 :

اجمع العناصر المتشابهة

= - 1 \cdot 1 + cos^{2} (x) cos^{2} (x)

1 \cdot 1 = 1

1 \cdot 1

1 \cdot 1 = 1 :

اضرب الأعداد

= 1

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

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

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

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

= cos^{2 + 2} (x)

2 + 2 = 4 :

اجمع الأعداد

= cos^{4} (x)

= - 1 + cos^{4} (x)

= - 1 + cos^{4} (x)

= - 1 + cos^{4} (x)

= cos^{4} (x) - 1

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e 1 + sec^{2} (x) sin (x) = sec^{2} (x)

p ro v e \frac{cot ( x )}{csc ( x ) - sin ( x )} = sec (x)

p ro v e sin (x) = 5 cos (x)

p ro v e \frac{sin ( 3 x ) + sin ( 5 x )}{cos ( 3 x ) - cos ( 5 x )} = cot (x)

p ro v e \frac{1}{sin ( x ) - sin ( x ) cos ( x )} = csc (x)