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prove ((sec(θ)-tan(θ))^2+1)/(csc(θ)(sec(θ)-tan(θ)))=2tan(θ)

الحلّ

prove

\frac{( sec ( θ ) - tan ( θ ) ) ^{2} + 1}{csc ( θ ) ( sec ( θ ) - tan ( θ ) )} = 2 tan (θ)

الحلّ

صحيح

خطوات الحلّ

\frac{( sec ( θ ) - tan ( θ ) ) ^{2} + 1}{csc ( θ ) ( sec ( θ ) - tan ( θ ) )} = 2 tan (θ)

قم بالعمل على الطرف الأيسر

\frac{( sec ( θ ) - tan ( θ ) ) ^{2} + 1}{csc ( θ ) ( sec ( θ ) - tan ( θ ) )}

sin, cos :

عبّر بواسطة

\frac{( sec ( θ ) - tan ( θ ) ) ^{2} + 1}{( sec ( θ ) - tan ( θ ) ) csc ( θ )}

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

:Use the basic trigonometric identity

= \frac{( \frac{1}{c o s ( θ )} - tan ( θ ) ) ^{2} + 1}{( \frac{1}{c o s ( θ )} - tan ( θ ) ) csc ( θ )}

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

:Use the basic trigonometric identity

= \frac{( \frac{1}{c o s ( θ )} - \frac{s i n ( θ )}{c o s ( θ )} ) ^{2} + 1}{( \frac{1}{c o s ( θ )} - \frac{s i n ( θ )}{c o s ( θ )} ) csc ( θ )}

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

وحّد الكسور:

\frac{1 - sin ( θ )}{cos ( θ )}

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

فعّل القانون

= \frac{1 - sin ( θ )}{cos ( θ )}

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

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

وحّد الكسور:

\frac{1 - sin ( θ )}{cos ( θ )}

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

فعّل القانون

= \frac{1 - sin ( θ )}{cos ( θ )}

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

(a) = a

:احذف الأقواس

= \frac{( \frac{1 - s i n ( θ )}{c o s ( θ )} ) ^{2} + 1}{\frac{1 - s i n ( θ )}{c o s ( θ )} \cdot \frac{1}{s i n ( θ )}}

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

اضرب بـ:

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

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

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

:اضرب كسور

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

(1 - sin (θ)) \cdot 1 = 1 - sin (θ)

(1 - sin (θ)) \cdot 1

(1 - sin (θ)) \cdot 1 = (1 - sin (θ)) :

اضرب

= (1 - sin (θ))

(a) = a

:احذف الأقواس

= 1 - sin (θ)

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

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

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

:فعّل قانون القوى

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

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

: استخدم ميزات الكسور التالية

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

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

وحّد:

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

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

1 \cdot cos^{2} (θ) = cos^{2} (θ) :

اضرب

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

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

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

اضرب بـ:

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

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

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

:اضرب كسور

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

cos (θ) :

إلغ العوامل المشتركة

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

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

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

: استخدم ميزات الكسور التالية

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

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

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

Rewrite using trig identities

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

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

:فعّل نطريّة فيتاغوروس

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

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

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

بسّط:

\frac{2 sin ( θ )}{cos ( θ )}

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

(1 - sin (θ))^{2} + 1 - sin^{2} (θ)

وسٌع:

- 2 sin (θ) + 2

(1 - sin (θ))^{2} + 1 - sin^{2} (θ)

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

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

:فعّل صيغة الضرب المختصر

a = 1, b = sin (θ)

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

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

بسّط:

1 - 2 sin (θ) + sin^{2} (θ)

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

1^{a} = 1

فعّل القانون

1^{2} = 1

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

2 \cdot 1 = 2 :

اضرب الأعداد

= 1 - 2 sin (θ) + sin^{2} (θ)

= 1 - 2 sin (θ) + sin^{2} (θ)

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

1 - 2 sin (θ) + sin^{2} (θ) + 1 - sin^{2} (θ)

بسّط:

- 2 sin (θ) + 2

1 - 2 sin (θ) + sin^{2} (θ) + 1 - sin^{2} (θ)

جمّع التعابير المتشابهة

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

sin^{2} (θ) - sin^{2} (θ) = 0 :

اجمع العناصر المتشابهة

= - 2 sin (θ) + 1 + 1

1 + 1 = 2 :

اجمع الأعداد

= - 2 sin (θ) + 2

= - 2 sin (θ) + 2

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

- 2 sin (θ) + 2

حلل إلى عوامل:

2 (- sin (θ) + 1)

- 2 sin (θ) + 2

أعد الكتابة كـ

= - 2 sin (θ) + 2 \cdot 1

2

قم باخراج العامل المشترك

= 2 (- sin (θ) + 1)

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

- sin (θ) + 1 :

إلغ العوامل المشتركة

= \frac{2 sin ( θ )}{cos ( θ )}

= \frac{2 sin ( θ )}{cos ( θ )}

= \frac{2 sin ( θ )}{cos ( θ )}

Rewrite using trig identities

= 2 \cdot \frac{sin ( θ )}{cos ( θ )}

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

:Use the basic trigonometric identity

2 tan (θ)

2 tan (θ)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

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