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prove (cos(x))/(csc(x)+1)+(cos(x))/(csc(x)-1)=2tan(x)

الحلّ

prove

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

الحلّ

صحيح

خطوات الحلّ

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

قم بالعمل على الطرف الأيسر

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

sin, cos :

عبّر بواسطة

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

csc (x) = \frac{1}{sin ( x )}

:Use the basic trigonometric identity

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

\frac{cos ( x )}{- 1 + \frac{1}{s i n ( x )}} + \frac{cos ( x )}{1 + \frac{1}{s i n ( x )}}

بسّط:

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

\frac{cos ( x )}{- 1 + \frac{1}{s i n ( x )}} + \frac{cos ( x )}{1 + \frac{1}{s i n ( x )}}

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

\frac{cos ( x )}{- 1 + \frac{1}{s i n ( x )}}

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

وحّد:

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

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

اضرب

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

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

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

: استخدم ميزات الكسور التالية

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

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

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

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

وحّد:

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

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

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

:حوّل الأعداد لكسور

= \frac{1 \cdot sin ( x )}{sin ( x )} + \frac{1}{sin ( x )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{1 \cdot sin ( x ) + 1}{sin ( x )}

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

اضرب

= \frac{sin ( x ) + 1}{sin ( x )}

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

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

: استخدم ميزات الكسور التالية

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

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

- sin (x) + 1, sin (x) + 1

المضاعف المشترك الأصغر لـ:

(sin (x) + 1) (- sin (x) + 1)

- sin (x) + 1, sin (x) + 1

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

- sin (x) + 1

sin (x) + 1

= (sin (x) + 1) (- sin (x) + 1)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

(sin (x) + 1) (- sin (x) + 1)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

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

multiply the denominator and numerator by

sin (x) + 1

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

For

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

multiply the denominator and numerator by

- sin (x) + 1

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

وسٌع:

2 cos (x) sin (x)

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

cos (x) sin (x) (sin (x) + 1)

وسٌع:

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

cos (x) sin (x) (sin (x) + 1)

a (b + c) = ab + a c

: افتح أقواس بالاستعانة بـ

a = cos (x) sin (x), b = sin (x), c = 1

= cos (x) sin (x) sin (x) + cos (x) sin (x) \cdot 1

= cos (x) sin (x) sin (x) + 1 \cdot cos (x) sin (x)

cos (x) sin (x) sin (x) + 1 \cdot cos (x) sin (x)

بسّط:

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

cos (x) sin (x) sin (x) + 1 \cdot cos (x) sin (x)

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

cos (x) sin (x) sin (x)

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

:فعّل قانون القوى

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

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

1 + 1 = 2 :

اجمع الأعداد

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

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

1 \cdot cos (x) sin (x)

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

اضرب

= cos (x) sin (x)

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

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

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

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

وسٌع:

- sin^{2} (x) cos (x) + cos (x) sin (x)

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

a (b + c) = ab + a c

: افتح أقواس بالاستعانة بـ

a = cos (x) sin (x), b = - sin (x), c = 1

= cos (x) sin (x) (- sin (x)) + cos (x) sin (x) \cdot 1

فعّل قوانين سالب-موجب

+ (- a) = - a

= - cos (x) sin (x) sin (x) + 1 \cdot cos (x) sin (x)

- cos (x) sin (x) sin (x) + 1 \cdot cos (x) sin (x)

بسّط:

- sin^{2} (x) cos (x) + cos (x) sin (x)

- cos (x) sin (x) sin (x) + 1 \cdot cos (x) sin (x)

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

cos (x) sin (x) sin (x)

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

:فعّل قانون القوى

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

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

1 + 1 = 2 :

اجمع الأعداد

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

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

1 \cdot cos (x) sin (x)

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

اضرب

= cos (x) sin (x)

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

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

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

sin^{2} (x) cos (x) + cos (x) sin (x) - sin^{2} (x) cos (x) + cos (x) sin (x)

بسّط:

2 cos (x) sin (x)

sin^{2} (x) cos (x) + cos (x) sin (x) - sin^{2} (x) cos (x) + cos (x) sin (x)

sin^{2} (x) cos (x) - sin^{2} (x) cos (x) = 0 :

اجمع العناصر المتشابهة

= cos (x) sin (x) + cos (x) sin (x)

cos (x) sin (x) + cos (x) sin (x) = 2 cos (x) sin (x) :

اجمع العناصر المتشابهة

= 2 cos (x) sin (x)

= 2 cos (x) sin (x)

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

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

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

Rewrite using trig identities

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

(1 + sin (x)) (1 - sin (x))

وسٌع:

1 - sin^{2} (x)

(1 + sin (x)) (1 - sin (x))

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

فعّل قانون فرق المربّعات

a = 1, b = sin (x)

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

1^{a} = 1

فعّل القانون

1^{2} = 1

= 1 - sin^{2} (x)

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

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

:فعّل نطريّة فيتاغوروس

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

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

cos (x) :

إلغ العوامل المشتركة

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

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

Rewrite using trig identities

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

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

:Use the basic trigonometric identity

2 tan (x)

2 tan (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

أمثلة شائعة

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