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شائع علم المثلثات >

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

الحلّ

prove

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

الحلّ

صحيح

خطوات الحلّ

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

قم بالعمل على الطرف الأيسر

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

1 - sin^{4} (x)

حلل إلى عوامل:

- (sin^{2} (x) + 1) (sin (x) + 1) (sin (x) - 1)

1 - sin^{4} (x)

- 1

قم باخراج العامل المشترك

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

sin^{4} (x) - 1

حلل إلى عوامل:

(sin^{2} (x) + 1) (sin (x) + 1) (sin (x) - 1)

sin^{4} (x) - 1

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

كـ

sin^{4} (x) - 1

اكتب مجددًا

sin^{4} (x) - 1

1^{2}

كـ

1

اكتب مجددًا

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

a^{b c} = (a^{b})^{c}

:فعّل قانون القوى

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

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

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

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

فعّل قانون فرق المربّعات

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

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

sin^{2} (x) - 1

حلل إلى عوامل:

(sin (x) + 1) (sin (x) - 1)

sin^{2} (x) - 1

1^{2}

كـ

1

اكتب مجددًا

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

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

فعّل قانون فرق المربّعات

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

= (sin (x) + 1) (sin (x) - 1)

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

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

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

Rewrite using trig identities

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

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

:فعّل نطريّة فيتاغوروس

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

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

بسّط

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

(sin (x) + 1) (sin (x) - 1)

وسٌع:

sin^{2} (x) - 1

(sin (x) + 1) (sin (x) - 1)

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

فعّل قانون فرق المربّعات

a = sin (x), b = 1

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

1^{a} = 1

فعّل القانون

1^{2} = 1

= sin^{2} (x) - 1

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

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

:فعّل نطريّة فيتاغوروس

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

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

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

بسّط:

2 cos^{2} (x)

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

- (- a) = a

فعّل القانون

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

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

وسٌع:

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

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

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

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

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

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

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)

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

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

cos^{4} (x) - cos^{4} (x) = 0 :

اجمع العناصر المتشابهة

= 2 cos^{2} (x)

= 2 cos^{2} (x)

= 2 cos^{2} (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

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