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العربية

prove (1-cos^2(x))(1+cot^2(x))=1

الحلّ

prove

(1 - cos^{2} (x)) (1 + cot^{2} (x)) = 1

صحيح

خطوات الحلّ

(1 - cos^{2} (x)) (1 + cot^{2} (x)) = 1

قم بالعمل على الطرف الأيسر

(1 - cos^{2} (x)) (1 + cot^{2} (x))

sin, cos :

عبّر بواسطة

(1 + cot^{2} (x)) (1 - cos^{2} (x))

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

:فعّل قانون القوى

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

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

وحّد:

\frac{sin ^{2} ( x ) + cos ^{2} ( x )}{sin ^{2} ( x )}

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

1 \cdot sin^{2} (x) = sin^{2} (x) :

اضرب

= \frac{sin ^{2} ( x ) + cos ^{2} ( x )}{sin ^{2} ( x )}

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

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

:اضرب كسور

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

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

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

Rewrite using trig identities

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

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

:فعّل نطريّة فيتاغوروس

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

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

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

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

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

(1 - cos^{2} (x)) (cos^{2} (x) + 1 - cos^{2} (x))

cos^{2} (x) + 1 - cos^{2} (x)

بسّط:

1

cos^{2} (x) + 1 - cos^{2} (x)

جمّع التعابير المتشابهة

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

cos^{2} (x) - cos^{2} (x) = 0 :

اجمع العناصر المتشابهة

= 1

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

(1 - cos^{2} (x)) \cdot 1 = (1 - cos^{2} (x)) :

اضرب

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

(a) = a

:احذف الأقواس

= 1 - cos^{2} (x)

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

\frac{a}{a} = 1

فعّل القانون

= 1

= 1

= 1

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e \frac{1 - cos ( - x )}{sec ( - x ) - 1} = cos (x)

p ro v e cos (3 θ) = 4 cos^{3} (θ) - 3 cos (θ)

p ro v e (sin (x) - cos (x))^{2} = 1 - sin (2 x)

p ro v e csc (θ) tan (θ) = sec (θ)

p ro v e \frac{1 + sin ( - x )}{1 + csc ( - x )} = - sin (x)