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شائع علم المثلثات >

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

الحلّ

prove

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

الحلّ

صحيح

خطوات الحلّ

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

قم بالعمل على الطرف الأيسر

\frac{cot ( θ ) - csc ( θ )}{cot ( θ ) + csc ( θ )}

sin, cos :

عبّر بواسطة

\frac{cot ( θ ) - csc ( θ )}{cot ( θ ) + csc ( θ )}

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

\frac{cos ( θ ) - 1}{cos ( θ ) + 1}

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

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

وحّد الكسور:

\frac{cos ( θ ) + 1}{sin ( θ )}

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

فعّل القانون

= \frac{cos ( θ ) + 1}{sin ( θ )}

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

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

وحّد الكسور:

\frac{cos ( θ ) - 1}{sin ( θ )}

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

فعّل القانون

= \frac{cos ( θ ) - 1}{sin ( θ )}

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

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

:اقسم الكسور

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

sin (θ) :

إلغ العوامل المشتركة

= \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

= \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

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

قم بالعمل على الطرف الأيمن

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

Rewrite using trig identities

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

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

:فعّل نطريّة فيتاغوروس

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

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

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

بسّط:

\frac{cos ( θ ) - 1}{cos ( θ ) + 1}

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

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

: استخدم ميزات الكسور التالية

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

(a) = a

:احذف الأقواس

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

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

اختزل:

- \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

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

1 - 2 cos (θ) + cos^{2} (θ)

حلل إلى عوامل:

(cos (θ) - 1)^{2}

1 - 2 cos (θ) + cos^{2} (θ)

a x^{2} + b x + c

اكتب بالصورة الاعتياديّة

= cos^{2} (θ) - 2 cos (θ) + 1

cos^{2} (θ) - 2 cos (θ) \cdot 1 + 1^{2}

كـ

cos^{2} (θ) - 2 cos (θ) + 1

اكتب مجددًا

cos^{2} (θ) - 2 cos (θ) + 1

1^{2}

كـ

1

اكتب مجددًا

= cos^{2} (θ) - 2 cos (θ) + 1^{2}

2 cos (θ) \cdot 1

كـ

2 cos (θ)

اكتب مجددًا

= cos^{2} (θ) - 2 cos (θ) \cdot 1 + 1^{2}

= cos^{2} (θ) - 2 cos (θ) \cdot 1 + 1^{2}

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

:فعّل صيغة الضرب المختصر

a = cos (θ), b = 1

= (cos (θ) - 1)^{2}

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

1 - cos^{2} (θ)

حلل إلى عوامل:

- (cos (θ) + 1) (cos (θ) - 1)

1 - cos^{2} (θ)

- 1

قم باخراج العامل المشترك

= - (cos^{2} (θ) - 1)

cos^{2} (θ) - 1

حلل إلى عوامل:

(cos (θ) + 1) (cos (θ) - 1)

cos^{2} (θ) - 1

1^{2}

كـ

1

اكتب مجددًا

= cos^{2} (θ) - 1^{2}

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

فعّل قانون فرق المربّعات

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

= (cos (θ) + 1) (cos (θ) - 1)

= - (cos (θ) + 1) (cos (θ) - 1)

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

cos (θ) - 1 :

إلغ العوامل المشتركة

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

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

- (- a) = a

فعّل القانون

= \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

= \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

= \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

أمثلة شائعة

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