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פּוֹפּוּלָרִי טריגונומטריה >

prove sec(x)-tan(x)(sin(x))= 1/(sec(x))

פתרון

prove

sec (x) - tan (x) (sin (x)) = \frac{1}{sec ( x )}

פתרון

נכון

צעדי פתרון

sec (x) - tan (x) sin (x) = \frac{1}{sec ( x )}

עבוד על אגף שמאל

sec (x) - tan (x) sin (x)

sin, cos :

בטא באמצאות

sec (x) - sin (x) tan (x)

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

:Use the basic trigonometric identity

= \frac{1}{cos ( x )} - sin (x) tan (x)

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

:Use the basic trigonometric identity

= \frac{1}{cos ( x )} - sin (x) \frac{sin ( x )}{cos ( x )}

\frac{1}{cos ( x )} - sin (x) \frac{sin ( x )}{cos ( x )}

פשט את:

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

\frac{1}{cos ( x )} - sin (x) \frac{sin ( x )}{cos ( x )}

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

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

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

:הכפל שברים

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

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

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

:הפעל את חוק החזקות

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

= sin^{1 + 1} (x)

1 + 1 = 2 :

חבר את המספרים

= sin^{2} (x)

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

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

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

הפעל את החוק

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

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

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

Rewrite using trig identities

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

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

:הפעל זהות פיטגורית

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

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

cos (x) :

בטל את הגורמים המשותפים

= cos (x)

= cos (x)

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

:Use the basic trigonometric identity

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

הראנו ששני האגפים יכולים להיות מאותה הצורה

\Rightarrow נכון

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