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العربية

prove 8cos^4(x)-8cos^2(x)+1=cos(4x)

الحلّ

prove

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

صحيح

خطوات الحلّ

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

قم بالعمل على الطرف الأيمن

cos (4 x)

Rewrite using trig identities

cos (4 x)

= cos (2 \cdot 2 x)

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

:فعّل متطابقة الزاوية المضاعفة

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

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

:فعّل متطابقة الزاوية المضاعفة

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

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

بسّط:

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

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

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

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

:فعّل صيغة الضرب المختصر

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

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

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

بسّط:

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

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

1^{a} = 1

فعّل القانون

1^{2} = 1

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

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

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

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

:فعّل قانون القوى

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

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

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

:فعّل قانون القوى

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

2 \cdot 2 = 4 :

اضرب الأعداد

= cos^{4} (x)

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

2^{2} = 4

= 4 cos^{4} (x)

2 \cdot 2 cos^{2} (x) \cdot 1 = 4 cos^{2} (x)

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

2 \cdot 2 \cdot 1 = 4 :

اضرب الأعداد

= 4 cos^{2} (x)

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

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

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

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

وسٌع:

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

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

فعّل قانون ضرب الأقواس

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

فعّل قوانين سالب-موجب

+ (- a) = - a

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

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

بسّط:

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

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

2 \cdot 4 = 8 :

اضرب الأعداد

= 8 cos^{4} (x) - 8 cos^{2} (x) + 2 \cdot 1

2 \cdot 1 = 2 :

اضرب الأعداد

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

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

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

2 - 1 = 1 :

اطرح/اجمع الأعداد

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

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

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

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e \frac{cos ( 2 x ) - cos ( 4 x )}{sin ( 2 x ) sin ( 4 x )} = tan (x)

p ro v e \frac{tan ( x ) + cot ( x )}{csc ( x )} = sec (x)

p ro v e tan (- x) = - tan (x)

p ro v e cot (θ) = \frac{sin ( 2 θ )}{1 - cos ( 2 θ )}

p ro v e (sec^{2} (x) - 1) cos (x) = tan (x) sin (x)