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prove csc^2(θ)tan^2(θ)-1=tan^2(θ)

פתרון

prove

csc^{2} (θ) tan^{2} (θ) - 1 = tan^{2} (θ)

נכון

צעדי פתרון

csc^{2} (θ) tan^{2} (θ) - 1 = tan^{2} (θ)

עבוד על אגף שמאל

csc^{2} (θ) tan^{2} (θ) - 1

sin, cos :

בטא באמצאות

- 1 + csc^{2} (θ) tan^{2} (θ)

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

פשט את:

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

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

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

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

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

(\frac{1}{sin ( θ )})^{2}

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

:הפעל את חוק החזקות

= \frac{1 ^{2}}{sin ^{2} ( θ )}

1^{a} = 1

הפעל את החוק

1^{2} = 1

= \frac{1}{sin ^{2} ( θ )}

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

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

(\frac{sin ( θ )}{cos ( θ )})^{2}

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

:הפעל את חוק החזקות

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

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

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

:הכפל שברים

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

sin^{2} (θ) :

בטל את הגורמים המשותפים

= \frac{1}{cos ^{2} ( θ )}

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

1 \cdot cos^{2} (θ) = cos^{2} (θ) :

הכפל

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

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

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

Rewrite using trig identities

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

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

:הפעל זהות פיטגורית

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

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

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

Rewrite using trig identities

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

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

:Use the basic trigonometric identity

\frac{sin ( θ ) tan ( θ )}{cos ( θ )}

= tan (θ) \frac{sin ( θ )}{cos ( θ )}

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

:Use the basic trigonometric identity

tan (θ) tan (θ)

tan (θ) tan (θ)

פשט את:

tan^{2} (θ)

tan (θ) tan (θ)

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

:הפעל את חוק החזקות

tan (θ) tan (θ) = tan^{1 + 1} (θ)

= tan^{1 + 1} (θ)

1 + 1 = 2 :

חבר את המספרים

= tan^{2} (θ)

tan^{2} (θ)

tan^{2} (θ)

הראנו ששני האגפים יכולים להיות מאותה הצורה

\Rightarrow נכון

p ro v e (1 - sin^{2} (x)) csc (x) = cos (x) cot (x)

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)