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prove (1+tan(x))/(1-tan(x))+(1+cot(x))/(1-cot(x))=0

الحلّ

prove

\frac{1 + tan ( x )}{1 - tan ( x )} + \frac{1 + cot ( x )}{1 - cot ( x )} = 0

الحلّ

صحيح

خطوات الحلّ

\frac{1 + tan ( x )}{1 - tan ( x )} + \frac{1 + cot ( x )}{1 - cot ( x )} = 0

قم بالعمل على الطرف الأيسر

\frac{1 + tan ( x )}{1 - tan ( x )} + \frac{1 + cot ( x )}{1 - cot ( x )}

sin, cos :

عبّر بواسطة

\frac{1 + cot ( x )}{1 - cot ( x )} + \frac{1 + tan ( x )}{1 - tan ( x )}

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

:Use the basic trigonometric identity

= \frac{1 + \frac{c o s ( x )}{s i n ( x )}}{1 - \frac{c o s ( x )}{s i n ( x )}} + \frac{1 + tan ( x )}{1 - tan ( x )}

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

:Use the basic trigonometric identity

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

\frac{1 + \frac{c o s ( x )}{s i n ( x )}}{1 - \frac{c o s ( x )}{s i n ( x )}} + \frac{1 + \frac{s i n ( x )}{c o s ( x )}}{1 - \frac{s i n ( x )}{c o s ( x )}} = 0

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

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

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

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

وحّد:

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

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

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

:حوّل الأعداد لكسور

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

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

اضرب

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

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

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

وحّد:

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

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

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

:حوّل الأعداد لكسور

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

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

اضرب

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

= \frac{\frac{s i n ( x ) + c o s ( x )}{s i n ( x )}}{\frac{s i n ( x ) - c o s ( x )}{s i n ( x )}}

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

:اقسم الكسور

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

sin (x) :

إلغ العوامل المشتركة

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

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

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

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

وحّد:

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

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

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

:حوّل الأعداد لكسور

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

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

اضرب

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

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

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

وحّد:

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

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

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

:حوّل الأعداد لكسور

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

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

اضرب

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

= \frac{\frac{c o s ( x ) + s i n ( x )}{c o s ( x )}}{\frac{c o s ( x ) - s i n ( x )}{c o s ( x )}}

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

:اقسم الكسور

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

cos (x) :

إلغ العوامل المشتركة

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

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

sin (x) - cos (x), cos (x) - sin (x)

المضاعف المشترك الأصغر لـ:

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

sin (x) - cos (x), cos (x) - sin (x)

Lowest Common Multiplier (LCM)

cos (x) - sin (x) = - (sin (x) - cos (x))

= sin (x) - cos (x), - (sin (x) - cos (x))

Compute an expression comprised of factors that appear either in

sin (x) - cos (x)

cos (x) - sin (x)

= - (sin (x) - cos (x))

اكتب مجددًا الكسور بحيث يكون المقام مشترك

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

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

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

multiply the denominator and numerator by

- 1

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

: استخدم ميزات الكسور التالية

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

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

اختزل:

0

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

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

حلل إلى عوامل:

0

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

أعد الكتابة كـ

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

(cos (x) + sin (x))

قم باخراج العامل المشترك

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

بسّط

= 0

= \frac{0}{( sin ( x ) - cos ( x ) )}

بسّط

= 0

= - 0

= 0

= 0

= 0

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

أمثلة شائعة

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