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العربية

prove cot^2(A)-cos^2(A)=cot^2(A)cos^2(A)

الحلّ

prove

cot^{2} (A) - cos^{2} (A) = cot^{2} (A) cos^{2} (A)

صحيح

خطوات الحلّ

cot^{2} (A) - cos^{2} (A) = cot^{2} (A) cos^{2} (A)

قم بالعمل على الطرف الأيسر

cot^{2} (A) - cos^{2} (A)

sin, cos :

عبّر بواسطة

- cos^{2} (A) + cot^{2} (A)

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

:فعّل قانون القوى

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

cos^{2} (A) = \frac{cos ^{2} ( A ) sin ^{2} ( A )}{sin ^{2} ( A )}

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

= \frac{cos ^{2} ( A ) - cos ^{2} ( A ) sin ^{2} ( A )}{sin ^{2} ( A )}

\frac{cos ^{2} ( A ) - cos ^{2} ( A ) sin ^{2} ( A )}{sin ^{2} ( A )}

حلل إلى عوامل:

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

\frac{cos ^{2} ( A ) - cos ^{2} ( A ) sin ^{2} ( A )}{sin ^{2} ( A )}

cos^{2} (A) - cos^{2} (A) sin^{2} (A)

حلل إلى عوامل:

cos^{2} (A) (1 - sin^{2} (A))

cos^{2} (A) - cos^{2} (A) sin^{2} (A)

cos^{2} (A)

قم باخراج العامل المشترك

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

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

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

Rewrite using trig identities

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

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

:فعّل نطريّة فيتاغوروس

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

= \frac{cos ^{2} ( A ) cos ^{2} ( A )}{sin ^{2} ( A )}

cos^{2} (A) cos^{2} (A) = cos^{4} (A)

cos^{2} (A) cos^{2} (A)

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

cos^{2} (A) cos^{2} (A) = cos^{2 + 2} (A)

= cos^{2 + 2} (A)

2 + 2 = 4 :

اجمع الأعداد

= cos^{4} (A)

= \frac{cos ^{4} ( A )}{sin ^{2} ( A )}

= \frac{cos ^{4} ( A )}{sin ^{2} ( A )}

Rewrite using trig identities

= \frac{cos ^{3} ( A )}{sin ( A )} \cdot \frac{cos ( A )}{sin ( A )}

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

:Use the basic trigonometric identity

\frac{cos ^{3} ( A ) cot ( A )}{sin ( A )}

= cos^{2} (A) cot (A) \frac{cos ( A )}{sin ( A )}

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

:Use the basic trigonometric identity

cos^{2} (A) cot (A) cot (A)

cos^{2} (A) cot (A) cot (A)

بسّط:

cos^{2} (A) cot^{2} (A)

cos^{2} (A) cot (A) cot (A)

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

cot (A) cot (A) = cot^{1 + 1} (A)

= cos^{2} (A) cot^{1 + 1} (A)

1 + 1 = 2 :

اجمع الأعداد

= cos^{2} (A) cot^{2} (A)

cos^{2} (A) cot^{2} (A)

cos^{2} (A) cot^{2} (A)

= cot^{2} (A) cos^{2} (A)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e cos^{2} (2 t) + sin^{2} (2 t) = 1

p ro v e tan (x + \frac{π}{3}) = \frac{tan ( x ) + 3}{1 - 3 tan ( x )}

p ro v e \frac{1 - 2 sin ^{2} ( x )}{1 - sin ^{2} ( x )} = 1 - tan (2 x)

p ro v e sin (x + y) - sin (x - y) = 2 sin (y) cos (x)

p ro v e sin (2 a) = 2 cos (a) sin (a)