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פּוֹפּוּלָרִי טריגונומטריה >

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

פתרון

prove

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

פתרון

נכון

צעדי פתרון

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

עבוד על אגף שמאל

cos (x) - csc (x) cot (x)

sin, cos :

בטא באמצאות

cos (x) - cot (x) csc (x)

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

פשט את:

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

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

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

\frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )}

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

:הכפל שברים

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

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

הכפל

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

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

sin (x) sin (x)

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

:הפעל את חוק החזקות

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

= sin^{1 + 1} (x)

1 + 1 = 2 :

חבר את המספרים

= sin^{2} (x)

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

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

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

:המר את המספרים לשברים

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

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

:מאחר והמכנים שווים, חבר את המונים

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

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

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

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

פרק לגורמים את:

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

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

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

פרק לגורמים את:

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

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

cos (x)

הוצא את הגורם המשותף

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

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

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

Rewrite using trig identities

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

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

:הפעל זהות פיטגורית

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

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

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

פשט את:

- \frac{cos ^{3} ( x )}{sin ^{2} ( x )}

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

(- a) = - a

:הסר סוגריים

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

- cos^{2} (x) cos (x) = - cos^{3} (x)

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

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

:הפעל את חוק החזקות

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

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

2 + 1 = 3 :

חבר את המספרים

= - cos^{3} (x)

= \frac{- cos ^{3} ( x )}{sin ^{2} ( x )}

\frac{- a}{b} = - \frac{a}{b}

: השתמש בתכונת השברים הבאה

= - \frac{cos ^{3} ( x )}{sin ^{2} ( x )}

= - \frac{cos ^{3} ( x )}{sin ^{2} ( x )}

= - \frac{cos ^{3} ( x )}{sin ^{2} ( x )}

Rewrite using trig identities

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

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

:Use the basic trigonometric identity

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

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

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

:Use the basic trigonometric identity

- cos (x) cot (x) cot (x)

- cos (x) cot (x) cot (x)

פשט את:

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

- cos (x) cot (x) cot (x)

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

:הפעל את חוק החזקות

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

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

1 + 1 = 2 :

חבר את המספרים

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

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

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

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

הראנו ששני האגפים יכולים להיות מאותה הצורה

\Rightarrow נכון

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