ar

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

شائع علم المثلثات >

prove (1-4sec^2(x))/(1+2sec(x))=1-2sec(x)

الحلّ

prove

\frac{1 - 4 sec ^{2} ( x )}{1 + 2 sec ( x )} = 1 - 2 sec (x)

الحلّ

صحيح

خطوات الحلّ

\frac{1 - 4 sec ^{2} ( x )}{1 + 2 sec ( x )} = 1 - 2 sec (x)

قم بالعمل على الطرف الأيسر

\frac{1 - 4 sec ^{2} ( x )}{1 + 2 sec ( x )}

\frac{1 - 4 sec ^{2} ( x )}{1 + 2 sec ( x )}

بسّط:

- 2 sec (x) + 1

\frac{1 - 4 sec ^{2} ( x )}{1 + 2 sec ( x )}

1 - 4 sec^{2} (x)

حلل إلى عوامل:

- (2 sec (x) + 1) (2 sec (x) - 1)

1 - 4 sec^{2} (x)

- 1

قم باخراج العامل المشترك

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

4 sec^{2} (x) - 1

حلل إلى عوامل:

(2 sec (x) + 1) (2 sec (x) - 1)

4 sec^{2} (x) - 1

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

كـ

4 sec^{2} (x) - 1

اكتب مجددًا

4 sec^{2} (x) - 1

2^{2}

كـ

4

اكتب مجددًا

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

1^{2}

كـ

1

اكتب مجددًا

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

a^{m} b^{m} = (ab)^{m}

:فعّل قانون القوى

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

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

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

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

فعّل قانون فرق المربّعات

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

= (2 sec (x) + 1) (2 sec (x) - 1)

= - (2 sec (x) + 1) (2 sec (x) - 1)

= - \frac{( 2 sec ( x ) + 1 ) ( 2 sec ( x ) - 1 )}{1 + 2 sec ( x )}

2 sec (x) + 1 :

إلغ العوامل المشتركة

= - (2 sec (x) - 1)

- (2 sec (x) - 1) = - 2 sec (x) + 1

اجعل القيمة سالبة لـ

= - 2 sec (x) + 1

= 1 - 2 sec (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

أمثلة شائعة

prove tan(x)=cot(pi/2-x)

p ro v e tan (x) = cot (\frac{π}{2} - x)

prove sec(2x)=(csc^2(x))/(csc^2(x)-2)

p ro v e sec (2 x) = \frac{csc ^{2} ( x )}{csc ^{2} ( x ) - 2}

prove (1-cos(a))/(1+cos(a))=tan^2(a/2)

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

prove csc^2(x)-cos(x)sec(x)=cot^2(x)

p ro v e csc^{2} (x) - cos (x) sec (x) = cot^{2} (x)

prove (csc(x)+1)/(cot(x)+cos(x))=sec(x)

p ro v e \frac{csc ( x ) + 1}{cot ( x ) + cos ( x )} = sec (x)