English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

شائع علم المثلثات >

prove (cos(A))/(1+sin(A))+(cos(A))/(1-sin(A))=2sec(A)

الحلّ

prove

\frac{cos ( A )}{1 + sin ( A )} + \frac{cos ( A )}{1 - sin ( A )} = 2 sec (A)

الحلّ

صحيح

خطوات الحلّ

\frac{cos ( A )}{1 + sin ( A )} + \frac{cos ( A )}{1 - sin ( A )} = 2 sec (A)

قم بالعمل على الطرف الأيسر

\frac{cos ( A )}{1 + sin ( A )} + \frac{cos ( A )}{1 - sin ( A )}

Rewrite using trig identities

\frac{cos ( A )}{1 + sin ( A )} + \frac{cos ( A )}{1 - sin ( A )}

\frac{1 + sin ( A )}{1 + sin ( A )}

اضرب بـ

= \frac{cos ( A )}{1 + sin ( A )} + \frac{cos ( A )}{\frac{( 1 - s i n ( A ) ) ( 1 + s i n ( A ) )}{1 + s i n ( A )}}

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

فعّل قانون فرق المربّعات

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

= \frac{cos ( A )}{1 + sin ( A )} + \frac{cos ( A )}{\frac{1 - s i n ^{2} ( A )}{1 + s i n ( A )}}

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

:فعّل نطريّة فيتاغوروس

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

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

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

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

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

: استخدم ميزات الكسور التالية

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

cos (A) :

إلغ العوامل المشتركة

= \frac{1 + sin ( A )}{cos ( A )}

= \frac{cos ( A )}{sin ( A ) + 1} + \frac{sin ( A ) + 1}{cos ( A )}

= \frac{cos ( A )}{sin ( A ) + 1} + \frac{sin ( A ) + 1}{cos ( A )}

\frac{1 + sin ( A )}{cos ( A )} + \frac{cos ( A )}{1 + sin ( A )}

بسّط:

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

\frac{1 + sin ( A )}{cos ( A )} + \frac{cos ( A )}{1 + sin ( A )}

cos (A), 1 + sin (A)

المضاعف المشترك الأصغر لـ:

cos (A) (sin (A) + 1)

cos (A), 1 + sin (A)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos (A)

1 + sin (A)

= cos (A) (sin (A) + 1)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

cos (A) (sin (A) + 1)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{1 + sin ( A )}{cos ( A )} :

multiply the denominator and numerator by

sin (A) + 1

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

For

\frac{cos ( A )}{1 + sin ( A )} :

multiply the denominator and numerator by

cos (A)

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

Rewrite using trig identities

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

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

:فعّل نطريّة فيتاغوروس

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

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

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

بسّط:

\frac{2}{cos ( A )}

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

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

وسٌع:

2 sin (A) + 2

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

(1 + sin (A))^{2} : 1 + 2 sin (A) + sin^{2} (A)

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

:فعّل صيغة الضرب المختصر

a = 1, b = sin (A)

= 1^{2} + 2 \cdot 1 \cdot sin (A) + sin^{2} (A)

1^{2} + 2 \cdot 1 \cdot sin (A) + sin^{2} (A)

بسّط:

1 + 2 sin (A) + sin^{2} (A)

1^{2} + 2 \cdot 1 \cdot sin (A) + sin^{2} (A)

1^{a} = 1

فعّل القانون

1^{2} = 1

= 1 + 2 \cdot 1 \cdot sin (A) + sin^{2} (A)

2 \cdot 1 = 2 :

اضرب الأعداد

= 1 + 2 sin (A) + sin^{2} (A)

= 1 + 2 sin (A) + sin^{2} (A)

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

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

بسّط:

2 sin (A) + 2

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

جمّع التعابير المتشابهة

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

sin^{2} (A) - sin^{2} (A) = 0 :

اجمع العناصر المتشابهة

= 2 sin (A) + 1 + 1

1 + 1 = 2 :

اجمع الأعداد

= 2 sin (A) + 2

= 2 sin (A) + 2

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

2 sin (A) + 2

حلل إلى عوامل:

2 (sin (A) + 1)

2 sin (A) + 2

أعد الكتابة كـ

= 2 sin (A) + 2 \cdot 1

2

قم باخراج العامل المشترك

= 2 (sin (A) + 1)

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

sin (A) + 1 :

إلغ العوامل المشتركة

= \frac{2}{cos ( A )}

= \frac{2}{cos ( A )}

= \frac{2}{cos ( A )}

Rewrite using trig identities

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

:Use the basic trigonometric identity

\frac{2}{\frac{1}{s e c ( A )}}

بسّط

\frac{2}{\frac{1}{s e c ( A )}}

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

: استخدم ميزات الكسور التالية

= \frac{2 sec ( A )}{1}

\frac{a}{1} = a

فعّل القانون

= 2 sec (A)

2 sec (A)

2 sec (A)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

أمثلة شائعة

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

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

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

p ro v e - tan (y) + sec (y) = \frac{cos ( y )}{1 + sin ( y )}

prove sin(2z)=2sin(z)cos(z)

p ro v e sin (2 z) = 2 sin (z) cos (z)

prove cot(x/2)=((1+cos(x)))/(sin(x))

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

prove 1/(tan(b))+tan(b)=sec(b)csc(b)

p ro v e \frac{1}{tan ( b )} + tan (b) = sec (b) csc (b)