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prove (csc^2(x))/(cot^2(x))=1+tan^2(x)

פתרון

prove

\frac{csc ^{2} ( x )}{cot ^{2} ( x )} = 1 + tan^{2} (x)

נכון

צעדי פתרון

\frac{csc ^{2} ( x )}{cot ^{2} ( x )} = 1 + tan^{2} (x)

עבוד על אגף שמאל

\frac{csc ^{2} ( x )}{cot ^{2} ( x )}

sin, cos :

בטא באמצאות

\frac{csc ^{2} ( x )}{cot ^{2} ( x )}

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

:Use the basic trigonometric identity

= \frac{( \frac{1}{s i n ( x )} ) ^{2}}{cot ^{2} ( x )}

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

:Use the basic trigonometric identity

= \frac{( \frac{1}{s i n ( x )} ) ^{2}}{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}

\frac{( \frac{1}{s i n ( x )} ) ^{2}}{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}

פשט את:

\frac{1}{cos ^{2} ( x )}

\frac{( \frac{1}{s i n ( x )} ) ^{2}}{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}

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

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

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

:הפעל את חוק החזקות

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

= \frac{( \frac{1}{s i n ( x )} ) ^{2}}{\frac{c o s ^{2} ( x )}{s i n ^{2} ( x )}}

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

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

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

:הפעל את חוק החזקות

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

1^{a} = 1

הפעל את החוק

1^{2} = 1

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

= \frac{\frac{1}{s i n ^{2} ( x )}}{\frac{c o s ^{2} ( x )}{s i n ^{2} ( x )}}

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

:חלק את השברים

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

פשט

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

sin^{2} (x) :

בטל את הגורמים המשותפים

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

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

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

עבוד על אגף ימין

1 + tan^{2} (x)

sin, cos :

בטא באמצאות

1 + tan^{2} (x)

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

:Use the basic trigonometric identity

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

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

פשט את:

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

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

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

:הפעל את חוק החזקות

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

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

:המר את המספרים לשברים

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

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

:מאחר והמכנים שווים, חבר את המונים

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

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

הכפל

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

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

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

Rewrite using trig identities

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

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

:הפעל זהות פיטגורית

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

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

הראנו ששני האגפים יכולים להיות מאותה הצורה

\Rightarrow נכון

p ro v e sin (\frac{x}{2}) = \frac{1 + cos ( x )}{2}

p ro v e 1 + \frac{1}{tan ^{2} ( θ )} = \frac{1}{sin ^{2} ( θ )}

p ro v e sin^{2} (x) \cdot (1 + cot^{2} (x)) = 1

p ro v e cos (x + 1) = cos (- x - 1)

p ro v e cot^{2} (x) + 1 = \frac{1}{sin ^{2} ( x )}