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פּוֹפּוּלָרִי טריגונומטריה >

prove (cos^4(x)-sin^4(x))/(cos^2(x))=1-tan^2(x)

פתרון

prove

\frac{cos ^{4} ( x ) - sin ^{4} ( x )}{cos ^{2} ( x )} = 1 - tan^{2} (x)

פתרון

נכון

צעדי פתרון

\frac{cos ^{4} ( x ) - sin ^{4} ( x )}{cos ^{2} ( x )} = 1 - tan^{2} (x)

עבוד על אגף שמאל

\frac{cos ^{4} ( x ) - sin ^{4} ( x )}{cos ^{2} ( x )}

Rewrite using trig identities

\frac{cos ^{4} ( x ) - sin ^{4} ( x )}{cos ^{2} ( x )}

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

:הפעל זהות פיטגורית

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

= \frac{cos ^{4} ( x ) - sin ^{4} ( x )}{1 - sin ^{2} ( x )}

= \frac{cos ^{4} ( x ) - sin ^{4} ( x )}{1 - sin ^{2} ( x )}

עבוד על אגף ימין

1 - tan^{2} (x)

Rewrite using trig identities

1 - tan^{2} (x)

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

:הפעל זהות פיטגורית

= cos^{2} (x) + sin^{2} (x) - tan^{2} (x)

= cos^{2} (x) + sin^{2} (x) - tan^{2} (x)

sin, cos :

בטא באמצאות

cos^{2} (x) + sin^{2} (x) - tan^{2} (x)

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

:Use the basic trigonometric identity

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

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

פשט את:

\frac{cos ^{4} ( x ) + sin ^{2} ( x ) cos ^{2} ( x ) - sin ^{2} ( x )}{cos ^{2} ( x )}

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

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

:הפעל את חוק החזקות

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

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

:המר את המספרים לשברים

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

cos^{2} (x) cos^{2} (x) + sin^{2} (x) cos^{2} (x) - sin^{2} (x) = cos^{4} (x) + sin^{2} (x) cos^{2} (x) - sin^{2} (x)

cos^{2} (x) cos^{2} (x) + sin^{2} (x) cos^{2} (x) - sin^{2} (x)

cos^{2} (x) cos^{2} (x) = cos^{4} (x)

cos^{2} (x) cos^{2} (x)

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

:הפעל את חוק החזקות

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

= cos^{2 + 2} (x)

2 + 2 = 4 :

חבר את המספרים

= cos^{4} (x)

= cos^{4} (x) + sin^{2} (x) cos^{2} (x) - sin^{2} (x)

= \frac{cos ^{4} ( x ) + sin ^{2} ( x ) cos ^{2} ( x ) - sin ^{2} ( x )}{cos ^{2} ( x )}

= \frac{cos ^{4} ( x ) + sin ^{2} ( x ) cos ^{2} ( x ) - sin ^{2} ( x )}{cos ^{2} ( x )}

= \frac{cos ^{4} ( x ) - sin ^{2} ( x ) + cos ^{2} ( x ) sin ^{2} ( x )}{cos ^{2} ( x )}

Rewrite using trig identities

\frac{cos ^{4} ( x ) - sin ^{2} ( x ) + cos ^{2} ( x ) sin ^{2} ( x )}{cos ^{2} ( x )}

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

:הפעל זהות פיטגורית

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

= \frac{cos ^{4} ( x ) - sin ^{2} ( x ) + ( 1 - sin ^{2} ( x ) ) sin ^{2} ( x )}{1 - sin ^{2} ( x )}

cos^{4} (x) - sin^{2} (x) + (1 - sin^{2} (x)) sin^{2} (x)

הרחב את:

cos^{4} (x) - sin^{4} (x)

cos^{4} (x) - sin^{2} (x) + (1 - sin^{2} (x)) sin^{2} (x)

= cos^{4} (x) - sin^{2} (x) + sin^{2} (x) (1 - sin^{2} (x))

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

הרחב את:

sin^{2} (x) - sin^{4} (x)

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

a (b - c) = ab - a c

: פתח סוגריים בעזרת

a = sin^{2} (x), b = 1, c = sin^{2} (x)

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

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

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

פשט את:

sin^{2} (x) - sin^{4} (x)

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

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

1 \cdot sin^{2} (x)

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

הכפל

= sin^{2} (x)

sin^{2} (x) sin^{2} (x) = sin^{4} (x)

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

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

:הפעל את חוק החזקות

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

= sin^{2 + 2} (x)

2 + 2 = 4 :

חבר את המספרים

= sin^{4} (x)

= sin^{2} (x) - sin^{4} (x)

= sin^{2} (x) - sin^{4} (x)

= cos^{4} (x) - sin^{2} (x) + sin^{2} (x) - sin^{4} (x)

- sin^{2} (x) + sin^{2} (x) = 0 :

חבר איברים דומים

= cos^{4} (x) - sin^{4} (x)

= \frac{cos ^{4} ( x ) - sin ^{4} ( x )}{1 - sin ^{2} ( x )}

= \frac{cos ^{4} ( x ) - sin ^{4} ( x )}{1 - sin ^{2} ( x )}

הראנו ששני האגפים יכולים להיות מאותה הצורה

\Rightarrow נכון

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