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פּוֹפּוּלָרִי >

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

פתרון

prove

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

נכון

צעדי פתרון

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

עבוד על אגף שמאל

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

sin, cos :

בטא באמצאות

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

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

:Use the basic trigonometric identity

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

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

פשט את:

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

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

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

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

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

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

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

:הפעל את חוק החזקות

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

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

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

:הכפל שברים

= \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}

:הפעל את חוק החזקות

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

= cos^{2 + 2} (A)

2 + 2 = 4 :

חבר את המספרים

= cos^{4} (A)

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

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

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

:המר את המספרים לשברים

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

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

:מאחר והמכנים שווים, חבר את המונים

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

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

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

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

פרק לגורמים את:

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

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

cos^{4} (A) + cos^{2} (A) sin^{2} (A)

פרק לגורמים את:

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

cos^{4} (A) + cos^{2} (A) sin^{2} (A)

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

:הפעל את חוק החזקות

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

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

cos^{2} (A)

הוצא את הגורם המשותף

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

Rewrite using trig identities

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

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

:הפעל זהות פיטגורית

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

פשט

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

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

Rewrite using trig identities

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

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

:Use the basic trigonometric identity

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

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

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

:Use the basic trigonometric identity

cot (A) cot (A)

cot (A) cot (A)

פשט את:

cot^{2} (A)

cot (A) cot (A)

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

:הפעל את חוק החזקות

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

= cot^{1 + 1} (A)

1 + 1 = 2 :

חבר את המספרים

= cot^{2} (A)

cot^{2} (A)

cot^{2} (A)

הראנו ששני האגפים יכולים להיות מאותה הצורה

\Rightarrow נכון

cot (9 0^{\circ}) = \frac{1}{tan ( 9 0 ^{\circ} )}

sin^{2} (θ) + 9 cos^{2} (θ) + 2 cos (θ) = 2

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

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

\frac{- 3 sin ^{2} ( 2 x )}{2} = \frac{3 cos ( 4 x )}{4}