English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

פּוֹפּוּלָרִי >

prove cot^2(θ)(1+tan^2(θ))=csc^2(θ)

פתרון

prove

cot^{2} (θ) (1 + tan^{2} (θ)) = csc^{2} (θ)

פתרון

נכון

צעדי פתרון

cot^{2} (θ) (1 + tan^{2} (θ)) = csc^{2} (θ)

עבוד על אגף שמאל

cot^{2} (θ) (1 + tan^{2} (θ))

sin, cos :

בטא באמצאות

(1 + tan^{2} (θ)) cot^{2} (θ)

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

:Use the basic trigonometric identity

= (1 + (\frac{sin ( θ )}{cos ( θ )})^{2}) cot^{2} (θ)

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

:Use the basic trigonometric identity

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

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

פשט את:

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

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

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

:הפעל את חוק החזקות

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

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

(\frac{cos ( θ )}{sin ( θ )})^{2}

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

:הפעל את חוק החזקות

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

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

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

:הכפל שברים

= \frac{cos ^{2} ( θ ) ( 1 + \frac{s i n ^{2} ( θ )}{c o s ^{2} ( θ )} )}{sin ^{2} ( θ )}

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

אחד את:

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

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

1 = \frac{1 cos ^{2} ( θ )}{cos ^{2} ( θ )}

:המר את המספרים לשברים

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

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

:מאחר והמכנים שווים, חבר את המונים

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

1 \cdot cos^{2} (θ) = cos^{2} (θ) :

הכפל

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

= \frac{\frac{c o s ^{2} ( θ ) + s i n ^{2} ( θ )}{c o s ^{2} ( θ )} cos ^{2} ( θ )}{sin ^{2} ( θ )}

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

הכפל ב:

cos^{2} (θ) + sin^{2} (θ)

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

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

:הכפל שברים

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

cos^{2} (θ) :

בטל את הגורמים המשותפים

= cos^{2} (θ) + sin^{2} (θ)

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

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

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

Rewrite using trig identities

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

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

:הפעל זהות פיטגורית

= \frac{1}{sin ^{2} ( θ )}

= \frac{1}{sin ^{2} ( θ )}

Rewrite using trig identities

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

:Use the basic trigonometric identity

\frac{1}{( \frac{1}{c s c ( θ )} ) ^{2}}

פשט

\frac{1}{( \frac{1}{c s c ( θ )} ) ^{2}}

(\frac{1}{csc ( θ )})^{2} = \frac{1}{csc ^{2} ( θ )}

(\frac{1}{csc ( θ )})^{2}

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

:הפעל את חוק החזקות

= \frac{1 ^{2}}{csc ^{2} ( θ )}

1^{a} = 1

הפעל את החוק

1^{2} = 1

= \frac{1}{csc ^{2} ( θ )}

= \frac{1}{\frac{1}{c s c ^{2} ( θ )}}

\frac{1}{\frac{b}{c}} = \frac{c}{b}

: השתמש בתכונת השברים הבאה

= \frac{csc ^{2} ( θ )}{1}

\frac{a}{1} = a

הפעל את החוק

= csc^{2} (θ)

csc^{2} (θ)

csc^{2} (θ)

הראנו ששני האגפים יכולים להיות מאותה הצורה

\Rightarrow נכון

דוגמאות פופולריות

prove sin(2A)=2sin(A)prove

sin (2 A) = 2 sin (A)

prove tan(θ)cos(θ)csc(θ)=1 prove

tan (θ) cos (θ) csc (θ) = 1

prove sin^2(x+pi/4)-cos^2(x-pi/4)=0 prove

sin^{2} (x + \frac{π}{4}) - cos^{2} (x - \frac{π}{4}) = 0

prove tan^2(u)-sin^2(u)=tan^2(u)sin^2(u)prove

tan^{2} (u) - sin^{2} (u) = tan^{2} (u) sin^{2} (u)

prove cos(x)-cos(x)sin^2(x)=cos^3(x)prove

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