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פּוֹפּוּלָרִי טריגונומטריה >

prove cos^2(2x)+4sin^2(x)cos^2(x)=1

פתרון

prove

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

פתרון

נכון

צעדי פתרון

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

עבוד על אגף שמאל

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

Rewrite using trig identities

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

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

:הפעל זהות של זווית כפולה

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

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

פשט את:

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

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

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

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

:הפעל נוסחת הכפל המקוצר

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

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

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

פשט את:

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

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

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

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

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

:הפעל את חוק החזקות

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

2 \cdot 2 = 4 :

הכפל את המספרים

= cos^{4} (x)

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

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

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

:הפעל את חוק החזקות

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

2 \cdot 2 = 4 :

הכפל את המספרים

= sin^{4} (x)

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

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

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

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

חבר איברים דומים

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

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

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

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

פרק לגורמים את:

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

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

חלק הביטוי לקבוצות

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

Rewrite in the form

a x^{2} + b x y + c y^{2}

Let

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

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

הגדרה

Factors of

1 : 1

1

Divisors (Factors)

Find the Prime factors of

1 :

Add 1

1

1

המחלקים של

1

For every two factors such that

u * v = 1,

check if

u + v = 2

Check

u = 1, v = 1 : u * v = 1, u + v = 2 \Rightarrow

נכון

u = 1, v = 1

Group into

(a x^{2} + ux y^{2}) + (v x^{2} y + c y^{2})

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

Reset

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

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

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

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

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

sin^{2} (x)

הוצא את הגורם

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

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

:הפעל את חוק החזקות

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

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

sin^{2} (x)

הוצא את הגורם המשותף

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

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

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

cos^{2} (x)

הוצא את הגורם

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

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

:הפעל את חוק החזקות

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

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

cos^{2} (x)

הוצא את הגורם המשותף

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

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

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

הוצא את הגורם המשותף

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

פשט

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

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

Rewrite using trig identities

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

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

:הפעל זהות פיטגורית

= 1^{2}

1^{a} = 1

הפעל את החוק

= 1

= 1

הראנו ששני האגפים יכולים להיות מאותה הצורה

\Rightarrow נכון

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