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

פתרון

prove

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

נכון

צעדי פתרון

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

עבוד על אגף שמאל

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

Rewrite using trig identities

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

\frac{1 + cos ( x )}{1 + cos ( x )}

הכפל ב

= \frac{sec ( x )}{\frac{( 1 - c o s ( x ) ) ( 1 + c o s ( x ) )}{1 + c o s ( x )}}

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

הפעל את חוק הפרש הריבועים

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

= \frac{sec ( x )}{\frac{1 - c o s ^{2} ( x )}{1 + c o s ( x )}}

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

:הפעל זהות פיטגורית

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

= \frac{sec ( x )}{\frac{s i n ^{2} ( x )}{1 + c o s ( x )}}

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

: השתמש בתכונת השברים הבאה

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

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

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

הרחב את:

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

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

(1 + cos (x)) sec (x)

הרחב את:

sec (x) + sec (x) cos (x)

(1 + cos (x)) sec (x)

= sec (x) (1 + cos (x))

a (b + c) = ab + a c

: פתח סוגריים בעזרת

a = sec (x), b = 1, c = cos (x)

= sec (x) \cdot 1 + sec (x) cos (x)

= 1 \cdot sec (x) + sec (x) cos (x)

1 \cdot sec (x) = sec (x) :

הכפל

= sec (x) + sec (x) cos (x)

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

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

: השתמש בתכונת השברים הבאה

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

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

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

Rewrite using trig identities

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

sec (x) cos (x) = 1

sec (x) cos (x)

sin, cos :

בטא באמצאות

sec (x) cos (x)

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

:Use the basic trigonometric identity

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

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

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

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

:הכפל שברים

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

cos (x) :

בטל את הגורמים המשותפים

= 1

= 1

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

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

הפעל את החוק

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

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

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

הראנו ששני האגפים יכולים להיות מאותה הצורה

\Rightarrow נכון

p ro v e tan (\frac{π}{4} - θ) = \frac{cos ( θ ) - sin ( θ )}{cos ( θ ) + sin ( θ )}

p ro v e cot (2 θ) = \frac{1}{2} sec (θ) csc (θ) - tan (θ)

p ro v e 8 cos^{2} (x) - 8 sin^{2} (x) = 8 - 16 sin^{2} (x)

p ro v e sec (x) - sin (x) \cdot tan (x) = cos (x)

p ro v e \frac{sec ( x ) csc ( x )}{cot ( x )} = sec^{2} (x)