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العربية

prove (1-sin^2(x))csc(x)=cos(x)cot(x)

الحلّ

prove

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

صحيح

خطوات الحلّ

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

قم بالعمل على الطرف الأيسر

(1 - sin^{2} (x)) csc (x)

sin, cos :

عبّر بواسطة

(1 - sin^{2} (x)) csc (x)

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

:اضرب كسور

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

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

1 \cdot (1 - sin^{2} (x))

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

اضرب

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

(a) = a

:احذف الأقواس

= 1 - sin^{2} (x)

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

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

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

Rewrite using trig identities

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

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

:فعّل نطريّة فيتاغوروس

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

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

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

Rewrite using trig identities

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

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

:Use the basic trigonometric identity

cos (x) cot (x)

cos (x) cot (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e sin (x) + sin (x) cot^{2} (x) = csc (x)

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

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

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

p ro v e csc (θ) cos (θ) = cot (θ)