English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

شائع علم المثلثات >

prove (sin(x)tan(x))/(cos(x)+1)=sec(x)-1

الحلّ

prove

\frac{sin ( x ) tan ( x )}{cos ( x ) + 1} = sec (x) - 1

الحلّ

صحيح

خطوات الحلّ

\frac{sin ( x ) tan ( x )}{cos ( x ) + 1} = sec (x) - 1

قم بالعمل على الطرف الأيسر

\frac{sin ( x ) tan ( x )}{cos ( x ) + 1}

sin, cos :

عبّر بواسطة

\frac{sin ( x ) tan ( x )}{1 + cos ( x )}

tan (x) = \frac{sin ( x )}{cos ( x )}

:Use the basic trigonometric identity

= \frac{sin ( x ) \frac{s i n ( x )}{c o s ( x )}}{1 + cos ( x )}

\frac{sin ( x ) \frac{s i n ( x )}{c o s ( x )}}{1 + cos ( x )}

بسّط:

\frac{sin ^{2} ( x )}{cos ( x ) ( 1 + cos ( x ) )}

\frac{sin ( x ) \frac{s i n ( x )}{c o s ( x )}}{1 + cos ( x )}

sin (x) \frac{sin ( x )}{cos ( x )}

اضرب بـ:

\frac{sin ^{2} ( x )}{cos ( x )}

sin (x) \frac{sin ( x )}{cos ( x )}

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

:اضرب كسور

= \frac{sin ( x ) sin ( x )}{cos ( x )}

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

sin (x) sin (x)

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

:فعّل قانون القوى

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

= sin^{1 + 1} (x)

1 + 1 = 2 :

اجمع الأعداد

= sin^{2} (x)

= \frac{sin ^{2} ( x )}{cos ( x )}

= \frac{\frac{s i n ^{2} ( x )}{c o s ( x )}}{1 + cos ( x )}

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

: استخدم ميزات الكسور التالية

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

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

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

Rewrite using trig identities

\frac{sin ^{2} ( x )}{( 1 + cos ( x ) ) cos ( x )}

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

:فعّل نطريّة فيتاغوروس

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

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

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

بسّط:

- \frac{cos ( x ) - 1}{cos ( x )}

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

1 - cos^{2} (x)

حلل إلى عوامل:

- (cos (x) + 1) (cos (x) - 1)

1 - cos^{2} (x)

- 1

قم باخراج العامل المشترك

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

cos^{2} (x) - 1

حلل إلى عوامل:

(cos (x) + 1) (cos (x) - 1)

cos^{2} (x) - 1

1^{2}

كـ

1

اكتب مجددًا

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

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

فعّل قانون فرق المربّعات

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

= (cos (x) + 1) (cos (x) - 1)

= - (cos (x) + 1) (cos (x) - 1)

= - \frac{( cos ( x ) + 1 ) ( cos ( x ) - 1 )}{( 1 + cos ( x ) ) cos ( x )}

cos (x) + 1 :

إلغ العوامل المشتركة

= - \frac{cos ( x ) - 1}{cos ( x )}

= - \frac{cos ( x ) - 1}{cos ( x )}

= - \frac{cos ( x ) - 1}{cos ( x )}

- \frac{- 1 + cos ( x )}{cos ( x )}

وسٌع:

\frac{1}{cos ( x )} - 1

- \frac{- 1 + cos ( x )}{cos ( x )}

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

: استخدم ميزات الكسور التالية

\frac{- 1 + cos ( x )}{cos ( x )} = - (- \frac{1}{cos ( x )}) - (\frac{cos ( x )}{cos ( x )})

= - (- \frac{1}{cos ( x )}) - (\frac{cos ( x )}{cos ( x )})

(a) = a, - (- a) = a

:احذف الأقواس

= \frac{1}{cos ( x )} - \frac{cos ( x )}{cos ( x )}

\frac{a}{a} = 1

فعّل القانون

= \frac{1}{cos ( x )} - 1

= - 1 + \frac{1}{cos ( x )}

- 1 + \frac{1}{cos ( x )}

بسّط:

\frac{- cos ( x ) + 1}{cos ( x )}

- 1 + \frac{1}{cos ( x )}

1 = \frac{1 cos ( x )}{cos ( x )}

:حوّل الأعداد لكسور

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

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

:بما أنّ المقامات متساوية، اجمع البسوط

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

1 \cdot cos (x) = cos (x) :

اضرب

= \frac{- cos ( x ) + 1}{cos ( x )}

= \frac{1 - cos ( x )}{cos ( x )}

Rewrite using trig identities

cos (x) = \frac{1}{sec ( x )}

:Use the basic trigonometric identity

\frac{1 - \frac{1}{s e c ( x )}}{\frac{1}{s e c ( x )}}

بسّط

\frac{1 - \frac{1}{s e c ( x )}}{\frac{1}{s e c ( x )}}

\frac{a}{\frac{b}{c}} = \frac{a \cdot c}{b}

: استخدم ميزات الكسور التالية

= \frac{( 1 - \frac{1}{s e c ( x )} ) sec ( x )}{1}

1 - \frac{1}{sec ( x )}

وحّد:

\frac{sec ( x ) - 1}{sec ( x )}

1 - \frac{1}{sec ( x )}

1 = \frac{1 sec ( x )}{sec ( x )}

:حوّل الأعداد لكسور

= \frac{1 \cdot sec ( x )}{sec ( x )} - \frac{1}{sec ( x )}

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{1 \cdot sec ( x ) - 1}{sec ( x )}

1 \cdot sec (x) = sec (x) :

اضرب

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

= \frac{\frac{s e c ( x ) - 1}{s e c ( x )} sec ( x )}{1}

\frac{a}{1} = a

: استخدم ميزات الكسور التالية

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

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

:اضرب كسور

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

sec (x) :

إلغ العوامل المشتركة

= sec (x) - 1

sec (x) - 1

sec (x) - 1

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

أمثلة شائعة

prove sin(a+b)-sin(a-b)=2sin(a)sin(b)

p ro v e sin (a + b) - sin (a - b) = 2 sin (a) sin (b)

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

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

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

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

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

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

prove sin^2(3x)=9sin^3(x)cos^3(x)

p ro v e sin^{2} (3 x) = 9 sin^{3} (x) cos^{3} (x)