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עברית

العربية

prove tan^2(u)-sin^2(u)=tan^2(u)sin^2(u)

פתרון

prove

tan^{2} (u) - sin^{2} (u) = tan^{2} (u) sin^{2} (u)

נכון

צעדי פתרון

tan^{2} (u) - sin^{2} (u) = tan^{2} (u) sin^{2} (u)

עבוד על אגף ימין

tan^{2} (u) sin^{2} (u)

Rewrite using trig identities

tan^{2} (u) sin^{2} (u)

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

:הפעל זהות פיטגורית

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

= tan^{2} (u) (1 - cos^{2} (u))

= tan^{2} (u) (1 - cos^{2} (u))

(1 - cos^{2} (u)) tan^{2} (u)

הרחב את:

tan^{2} (u) - tan^{2} (u) cos^{2} (u)

(1 - cos^{2} (u)) tan^{2} (u)

= tan^{2} (u) (1 - cos^{2} (u))

a (b - c) = ab - a c

: פתח סוגריים בעזרת

a = tan^{2} (u), b = 1, c = cos^{2} (u)

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

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

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

הכפל

= tan^{2} (u) - tan^{2} (u) cos^{2} (u)

= tan^{2} (u) - cos^{2} (u) tan^{2} (u)

Rewrite using trig identities

tan^{2} (u) - cos^{2} (u) tan^{2} (u)

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

:Use the basic trigonometric identity

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

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

פשט את:

sin^{2} (u)

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

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

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

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

:הפעל את חוק החזקות

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

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

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

:הכפל שברים

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

cos^{2} (u) :

בטל את הגורמים המשותפים

= sin^{2} (u)

= tan^{2} (u) - sin^{2} (u)

= tan^{2} (u) - sin^{2} (u)

הראנו ששני האגפים יכולים להיות מאותה הצורה

\Rightarrow נכון

p ro v e 1 - tan^{4} (A) = 2 sec^{2} (A) - sec^{4} (A)

p ro v e cos^{4} (θ) - sin^{4} (θ) = 2 cos^{2} (θ) - 1

p ro v e sin (x) tan (x) + cos (x) = \frac{1}{cos ( x )}

p ro v e \frac{1 - cos ( x )}{sin ( x )} = tan (\frac{x}{2})

p ro v e cos (4 θ) = 8 cos^{4} (θ) - 8 cos^{2} (θ) + 1