English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

شائع علم المثلثات >

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

الحلّ

prove

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

الحلّ

صحيح

خطوات الحلّ

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

قم بالعمل على الطرف الأيسر

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

sin, cos :

عبّر بواسطة

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

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

:Use the basic trigonometric identity

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

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

بسّط:

sin (A) + cos (A)

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

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

\frac{sin ( A )}{1 - \frac{c o s ( A )}{s i n ( A )}}

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

وحّد:

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

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

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

:حوّل الأعداد لكسور

= \frac{1 \cdot sin ( A )}{sin ( A )} - \frac{cos ( A )}{sin ( A )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{1 \cdot sin ( A ) - cos ( A )}{sin ( A )}

1 \cdot sin (A) = sin (A) :

اضرب

= \frac{sin ( A ) - cos ( A )}{sin ( A )}

= \frac{sin ( A )}{\frac{s i n ( A ) - c o s ( A )}{s i n ( A )}}

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

: استخدم ميزات الكسور التالية

= \frac{sin ( A ) sin ( A )}{sin ( A ) - cos ( A )}

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

sin (A) sin (A)

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

:فعّل قانون القوى

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

= sin^{1 + 1} (A)

1 + 1 = 2 :

اجمع الأعداد

= sin^{2} (A)

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

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

cos (A) - sin (A), sin (A) - cos (A)

المضاعف المشترك الأصغر لـ:

- (cos (A) - sin (A))

cos (A) - sin (A), sin (A) - cos (A)

Lowest Common Multiplier (LCM)

sin (A) - cos (A) = - (cos (A) - sin (A))

= cos (A) - sin (A), - (cos (A) - sin (A))

Compute an expression comprised of factors that appear either in

cos (A) - sin (A)

sin (A) - cos (A)

= - (cos (A) - sin (A))

اكتب مجددًا الكسور بحيث يكون المقام مشترك

- (cos (A) - sin (A))

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

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

multiply the denominator and numerator by

- 1

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

\frac{a}{- b} = - \frac{a}{b}

: استخدم ميزات الكسور التالية

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

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

اختزل:

- sin (A) - cos (A)

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

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

فعّل قانون فرق المربّعات

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

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

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

اختزل:

- (sin (A) + cos (A))

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

cos (A) - sin (A) = - (sin (A) - cos (A))

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

بسّط

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

sin (A) - cos (A) :

إلغ العوامل المشتركة

= - (sin (A) + cos (A))

= - (sin (A) + cos (A))

- (sin (A) + cos (A)) = - sin (A) - cos (A)

اجعل القيمة سالبة لـ

= - sin (A) - cos (A)

= - (- sin (A) - cos (A))

افتح أقواس

= - (- sin (A)) - (- cos (A))

فعّل قوانين سالب-موجب

- (- a) = a

= sin (A) + cos (A)

= sin (A) + cos (A)

= cos (A) + sin (A)

= sin (A) + cos (A)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

أمثلة شائعة

prove cos(x-pi/6)-sin(x+pi/3)=0

p ro v e cos (x - \frac{π}{6}) - sin (x + \frac{π}{3}) = 0

prove cot^2(α)+1= 1/(sin^2(α))

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

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

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

prove (cos(3x)-cos(5x))/(sin(3x)+sin(5x))=tan(x)

p ro v e \frac{cos ( 3 x ) - cos ( 5 x )}{sin ( 3 x ) + sin ( 5 x )} = tan (x)

prove cos^2(x)= 1/2+1/2 cos(2x)

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