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شائع علم المثلثات >

prove csc^2(x/2)=(2sec(x))/(sec(x)-1)

الحلّ

prove

csc^{2} (\frac{x}{2}) = \frac{2 sec ( x )}{sec ( x ) - 1}

الحلّ

صحيح

خطوات الحلّ

csc^{2} (\frac{x}{2}) = \frac{2 sec ( x )}{sec ( x ) - 1}

u = \frac{x}{2} :

على افتراض أنّ

csc^{2} (u) = \frac{2 sec ( 2 u )}{sec ( 2 u ) - 1}

csc^{2} (u) = \frac{2 sec ( 2 u )}{sec ( 2 u ) - 1}

أثبت أنّ:

صحيح

csc^{2} (u) = \frac{2 sec ( 2 u )}{sec ( 2 u ) - 1}

قم بالعمل على الطرف الأيمن

\frac{2 sec ( 2 u )}{sec ( 2 u ) - 1}

sin, cos :

عبّر بواسطة

\frac{2 sec ( 2 u )}{- 1 + sec ( 2 u )}

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

:Use the basic trigonometric identity

= \frac{2 \cdot \frac{1}{c o s ( 2 u )}}{- 1 + \frac{1}{c o s ( 2 u )}}

\frac{2 \cdot \frac{1}{c o s ( 2 u )}}{- 1 + \frac{1}{c o s ( 2 u )}}

بسّط:

\frac{2}{- cos ( 2 u ) + 1}

\frac{2 \cdot \frac{1}{c o s ( 2 u )}}{- 1 + \frac{1}{c o s ( 2 u )}}

- 1 + \frac{1}{cos ( 2 u )}

وحّد:

\frac{- cos ( 2 u ) + 1}{cos ( 2 u )}

- 1 + \frac{1}{cos ( 2 u )}

1 = \frac{1 cos ( 2 u )}{cos ( 2 u )}

:حوّل الأعداد لكسور

= - \frac{1 \cdot cos ( 2 u )}{cos ( 2 u )} + \frac{1}{cos ( 2 u )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{- 1 \cdot cos ( 2 u ) + 1}{cos ( 2 u )}

1 \cdot cos (2 u) = cos (2 u) :

اضرب

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

= \frac{2 \cdot \frac{1}{c o s ( 2 u )}}{\frac{- c o s ( 2 u ) + 1}{c o s ( 2 u )}}

2 \cdot \frac{1}{cos ( 2 u )}

اضرب بـ:

\frac{2}{cos ( 2 u )}

2 \cdot \frac{1}{cos ( 2 u )}

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

:اضرب كسور

= \frac{1 \cdot 2}{cos ( 2 u )}

1 \cdot 2 = 2 :

اضرب الأعداد

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

= \frac{\frac{2}{c o s ( 2 u )}}{\frac{- c o s ( 2 u ) + 1}{c o s ( 2 u )}}

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

:اقسم الكسور

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

cos (2 u) :

إلغ العوامل المشتركة

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

= \frac{2}{1 - cos ( 2 u )}

= \frac{2}{1 - cos ( 2 u )}

Rewrite using trig identities

\frac{2}{1 - cos ( 2 u )}

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

:فعّل متطابقة الزاوية المضاعفة

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

\frac{2}{1 - ( 1 - 2 sin ^{2} ( u ) )}

بسّط:

\frac{1}{sin ^{2} ( u )}

\frac{2}{1 - ( 1 - 2 sin ^{2} ( u ) )}

1 - (1 - 2 sin^{2} (u))

وسٌع:

2 sin^{2} (u)

1 - (1 - 2 sin^{2} (u))

- (1 - 2 sin^{2} (u)) : - 1 + 2 sin^{2} (u)

- (1 - 2 sin^{2} (u))

افتح أقواس

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

فعّل قوانين سالب-موجب

- (- a) = a, - (a) = - a

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

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

1 - 1 = 0

= 2 sin^{2} (u)

= \frac{2}{2 sin ^{2} ( u )}

\frac{2}{2} = 1 :

اقسم الأعداد

= \frac{1}{sin ^{2} ( u )}

= \frac{1}{sin ^{2} ( u )}

= \frac{1}{sin ^{2} ( u )}

Rewrite using trig identities

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

:Use the basic trigonometric identity

\frac{1}{( \frac{1}{c s c ( u )} ) ^{2}}

بسّط

\frac{1}{( \frac{1}{c s c ( u )} ) ^{2}}

(\frac{1}{csc ( u )})^{2} = \frac{1}{csc ^{2} ( u )}

(\frac{1}{csc ( u )})^{2}

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

:فعّل قانون القوى

= \frac{1 ^{2}}{csc ^{2} ( u )}

1^{a} = 1

فعّل القانون

1^{2} = 1

= \frac{1}{csc ^{2} ( u )}

= \frac{1}{\frac{1}{c s c ^{2} ( u )}}

\frac{1}{\frac{b}{c}} = \frac{c}{b}

: استخدم ميزات الكسور التالية

= \frac{csc ^{2} ( u )}{1}

\frac{a}{1} = a

فعّل القانون

= csc^{2} (u)

csc^{2} (u)

csc^{2} (u)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

csc^{2} (\frac{x}{2}) = \frac{2 sec ( x )}{sec ( x ) - 1}

لذلك

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

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