he

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

פּוֹפּוּלָרִי >

prove tan^4(x)-sec^4(x)=-2sec^2(x)+1

פתרון

prove

tan^{4} (x) - sec^{4} (x) = - 2 sec^{2} (x) + 1

פתרון

נכון

צעדי פתרון

tan^{4} (x) - sec^{4} (x) = - 2 sec^{2} (x) + 1

עבוד על אגף שמאל

tan^{4} (x) - sec^{4} (x)

- sec^{4} (x) + tan^{4} (x)

פרק לגורמים את:

(tan^{2} (x) + sec^{2} (x)) (tan (x) + sec (x)) (tan (x) - sec (x))

- sec^{4} (x) + tan^{4} (x)

(tan^{2} (x))^{2} - (sec^{2} (x))^{2}

בתור

tan^{4} (x) - sec^{4} (x)

כתוב מחדש את

tan^{4} (x) - sec^{4} (x)

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

:הפעל את חוק החזקות

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

= tan^{4} (x) - (sec^{2} (x))^{2}

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

:הפעל את חוק החזקות

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

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

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

x^{2} - y^{2} = (x + y) (x - y)

הפעל את חוק הפרש הריבועים

(tan^{2} (x))^{2} - (sec^{2} (x))^{2} = (tan^{2} (x) + sec^{2} (x)) (tan^{2} (x) - sec^{2} (x))

= (tan^{2} (x) + sec^{2} (x)) (tan^{2} (x) - sec^{2} (x))

tan^{2} (x) - sec^{2} (x)

פרק לגורמים את:

(tan (x) + sec (x)) (tan (x) - sec (x))

tan^{2} (x) - sec^{2} (x)

x^{2} - y^{2} = (x + y) (x - y)

הפעל את חוק הפרש הריבועים

tan^{2} (x) - sec^{2} (x) = (tan (x) + sec (x)) (tan (x) - sec (x))

= (tan (x) + sec (x)) (tan (x) - sec (x))

= (tan^{2} (x) + sec^{2} (x)) (tan (x) + sec (x)) (tan (x) - sec (x))

= (- sec (x) + tan (x)) (sec (x) + tan (x)) (sec^{2} (x) + tan^{2} (x))

Rewrite using trig identities

(- sec (x) + tan (x)) (sec (x) + tan (x)) (sec^{2} (x) + tan^{2} (x))

tan^{2} (x) + 1 = sec^{2} (x)

:הפעל זהות פיטגורית

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

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

sec^{2} (x) + sec^{2} (x) = 2 sec^{2} (x) :

חבר איברים דומים

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

(tan (x) + sec (x)) (tan (x) - sec (x))

הרחב את:

tan^{2} (x) - sec^{2} (x)

(tan (x) + sec (x)) (tan (x) - sec (x))

(a + b) (a - b) = a^{2} - b^{2}

הפעל את חוק הפרש הריבועים

a = tan (x), b = sec (x)

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

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

sec^{2} (x) = tan^{2} (x) + 1

:הפעל זהות פיטגורית

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

= (- 1 + 2 sec^{2} (x)) (- 1)

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

פשט את:

1 - 2 sec^{2} (x)

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

(- a) = - a

:הסר סוגריים

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

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

הכפל

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

פתח סוגריים

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

הפעל חוקי מינוס-פלוס

- (- a) = a, - (a) = - a

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

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

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

= - 2 sec^{2} (x) + 1

הראנו ששני האגפים יכולים להיות מאותה הצורה

\Rightarrow נכון

דוגמאות פופולריות

prove (tan(x)+cot(x))tan(x)=sec^2(x)prove

(tan (x) + cot (x)) tan (x) = sec^{2} (x)

prove (cos(x)+sin(x))^2-2sin(x)cos(x)=1 prove

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

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

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

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

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

prove 8csc^2(x)-3cot^2(x)=3+5csc^2(x)prove

8 csc^{2} (x) - 3 cot^{2} (x) = 3 + 5 csc^{2} (x)