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شائع علم المثلثات >

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

الحلّ

prove

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

الحلّ

صحيح

خطوات الحلّ

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

قم بالعمل على الطرف الأيسر

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

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

وسٌع:

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

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

- 1 + sec (x), 1 + sec (x)

المضاعف المشترك الأصغر لـ:

(sec (x) + 1) (sec (x) - 1)

- 1 + sec (x), 1 + sec (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

- 1 + sec (x)

1 + sec (x)

= (sec (x) + 1) (sec (x) - 1)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

(sec (x) + 1) (sec (x) - 1)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

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

multiply the denominator and numerator by

sec (x) + 1

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

For

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

multiply the denominator and numerator by

sec (x) - 1

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

(sec (x) + 1) (sec (x) - 1)

وسٌع:

sec^{2} (x) - 1

(sec (x) + 1) (sec (x) - 1)

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

فعّل قانون فرق المربّعات

a = sec (x), b = 1

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

1^{a} = 1

فعّل القانون

1^{2} = 1

= sec^{2} (x) - 1

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

cos (x) (sec (x) + 1) - cos (x) (sec (x) - 1)

وسٌع:

2 cos (x)

cos (x) (sec (x) + 1) - cos (x) (sec (x) - 1)

cos (x) (sec (x) + 1)

وسٌع:

cos (x) sec (x) + cos (x)

cos (x) (sec (x) + 1)

a (b + c) = ab + a c

: افتح أقواس بالاستعانة بـ

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

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

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

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

اضرب

= cos (x) sec (x) + cos (x)

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

- cos (x) (sec (x) - 1)

وسٌع:

- cos (x) sec (x) + cos (x)

- cos (x) (sec (x) - 1)

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

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

= - cos (x) sec (x) - (- cos (x)) \cdot 1

فعّل قوانين سالب-موجب

- (- a) = a

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

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

اضرب

= - cos (x) sec (x) + cos (x)

= cos (x) sec (x) + cos (x) - cos (x) sec (x) + cos (x)

cos (x) sec (x) + cos (x) - cos (x) sec (x) + cos (x)

بسّط:

2 cos (x)

cos (x) sec (x) + cos (x) - cos (x) sec (x) + cos (x)

cos (x) sec (x) - cos (x) sec (x) = 0 :

اجمع العناصر المتشابهة

= cos (x) + cos (x)

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

اجمع العناصر المتشابهة

= 2 cos (x)

= 2 cos (x)

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

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

قم بالعمل على الطرف الأيمن

\frac{2 cos ( x )}{tan ^{2} ( x )}

Rewrite using trig identities

\frac{2 cos ( x )}{tan ^{2} ( x )}

tan^{2} (x) + 1 = sec^{2} (x)

:فعّل نطريّة فيتاغوروس

tan^{2} (x) = sec^{2} (x) - 1

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

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

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

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