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פּוֹפּוּלָרִי >

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

פתרון

prove

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

פתרון

נכון

צעדי פתרון

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

עבוד על אגף שמאל

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

sin, cos :

בטא באמצאות

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

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

פשט את:

cos^{2} (x)

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

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

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

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

:חלק את השברים

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

פשט

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

sin^{2} (x) :

בטל את הגורמים המשותפים

= cos^{2} (x)

= cos^{2} (x)

= cos^{2} (x)

הראנו ששני האגפים יכולים להיות מאותה הצורה

\Rightarrow נכון

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