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פּוֹפּוּלָרִי >

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

פתרון

prove

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

פתרון

נכון

צעדי פתרון

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

עבוד על אגף ימין

(tan^{2} (x) + 1) (cos^{2} (x) - 1)

(- 1 + cos^{2} (x)) (1 + tan^{2} (x))

הרחב את:

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

(- 1 + cos^{2} (x)) (1 + tan^{2} (x))

(a + b) (c + d) = a c + a d + b c + b d

:

(a + b) (c + d) = a c + a d + b c + b d

הפעל חוק הפילוג

a = - 1, b = cos^{2} (x), c = 1, d = tan^{2} (x)

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

הפעל חוקי מינוס-פלוס

+ (- a) = - a

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

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

פשט את:

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

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

1 \cdot 1 = 1 :

הכפל את המספרים

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

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

הכפל

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

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

הכפל

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

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

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

Rewrite using trig identities

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

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

:הפעל זהות פיטגורית

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

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

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

:Use the basic trigonometric identity

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

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

פשט את:

sin^{2} (x)

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

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

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

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

:הפעל את חוק החזקות

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

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

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

:הכפל שברים

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

cos^{2} (x) :

בטל את הגורמים המשותפים

= sin^{2} (x)

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

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

חבר איברים דומים

= - tan^{2} (x)

= - tan^{2} (x)

הראנו ששני האגפים יכולים להיות מאותה הצורה

\Rightarrow נכון

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