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פּוֹפּוּלָרִי טריגונומטריה >

prove (1-cos^2(x))sec^2(x)=tan^2(x)

פתרון

prove

(1 - cos^{2} (x)) sec^{2} (x) = tan^{2} (x)

פתרון

נכון

צעדי פתרון

(1 - cos^{2} (x)) sec^{2} (x) = tan^{2} (x)

עבוד על אגף שמאל

(1 - cos^{2} (x)) sec^{2} (x)

sin, cos :

בטא באמצאות

(1 - cos^{2} (x)) sec^{2} (x)

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

:Use the basic trigonometric identity

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

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

פשט את:

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

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

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

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

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

:הפעל את חוק החזקות

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

1^{a} = 1

הפעל את החוק

1^{2} = 1

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

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

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

:הכפל שברים

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

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

1 \cdot (1 - cos^{2} (x))

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

הכפל

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

(a) = a

:הסר סוגריים

= 1 - cos^{2} (x)

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

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

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

Rewrite using trig identities

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

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

:הפעל זהות פיטגורית

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

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

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

Rewrite using trig identities

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

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

:Use the basic trigonometric identity

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

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

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

:Use the basic trigonometric identity

tan (x) tan (x)

tan (x) tan (x)

פשט את:

tan^{2} (x)

tan (x) tan (x)

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

:הפעל את חוק החזקות

tan (x) tan (x) = tan^{1 + 1} (x)

= tan^{1 + 1} (x)

1 + 1 = 2 :

חבר את המספרים

= tan^{2} (x)

tan^{2} (x)

tan^{2} (x)

הראנו ששני האגפים יכולים להיות מאותה הצורה

\Rightarrow נכון

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