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פּוֹפּוּלָרִי טריגונומטריה >

prove (tan^2(θ))/(sec(θ))=sin(θ)tan(θ)

פתרון

prove

\frac{tan ^{2} ( θ )}{sec ( θ )} = sin (θ) tan (θ)

פתרון

נכון

צעדי פתרון

\frac{tan ^{2} ( θ )}{sec ( θ )} = sin (θ) tan (θ)

עבוד על אגף שמאל

\frac{tan ^{2} ( θ )}{sec ( θ )}

sin, cos :

בטא באמצאות

\frac{tan ^{2} ( θ )}{sec ( θ )}

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

:Use the basic trigonometric identity

= \frac{( \frac{s i n ( θ )}{c o s ( θ )} ) ^{2}}{sec ( θ )}

sec (x) = \frac{1}{cos ( x )}

:Use the basic trigonometric identity

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

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

פשט את:

\frac{sin ^{2} ( θ )}{cos ( θ )}

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

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

: השתמש בתכונת השברים הבאה

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

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

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

הכפל ב:

\frac{sin ^{2} ( θ )}{cos ( θ )}

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

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

:הכפל שברים

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

cos (θ) :

בטל את הגורמים המשותפים

= \frac{sin ^{2} ( θ )}{cos ( θ )}

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

\frac{a}{1} = a

: השתמש בתכונת השברים הבאה

= \frac{sin ^{2} ( θ )}{cos ( θ )}

= \frac{sin ^{2} ( θ )}{cos ( θ )}

= \frac{sin ^{2} ( θ )}{cos ( θ )}

Rewrite using trig identities

= sin (θ) \frac{sin ( θ )}{cos ( θ )}

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

:Use the basic trigonometric identity

sin (θ) tan (θ)

sin (θ) tan (θ)

הראנו ששני האגפים יכולים להיות מאותה הצורה

\Rightarrow נכון

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