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Populaire Trigonométrie >

prouver (cot(θ)-csc(θ))/(cot(θ)+csc(θ))=(1-2cos(θ)+cos^2(θ))/(-sin^2(θ))

Solution

prouver

\frac{cot ( θ ) - csc ( θ )}{cot ( θ ) + csc ( θ )} = \frac{1 - 2 cos ( θ ) + cos ^{2} ( θ )}{- sin ^{2} ( θ )}

Solution

vrai

étapes des solutions

\frac{cot ( θ ) - csc ( θ )}{cot ( θ ) + csc ( θ )} = \frac{1 - 2 cos ( θ ) + cos ^{2} ( θ )}{- sin ^{2} ( θ )}

En manipulant le côté gauche

\frac{cot ( θ ) - csc ( θ )}{cot ( θ ) + csc ( θ )}

Exprimer avec sinus, cosinus

\frac{cot ( θ ) - csc ( θ )}{cot ( θ ) + csc ( θ )}

Utiliser l'identité trigonométrique de base:

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

= \frac{\frac{c o s ( θ )}{s i n ( θ )} - csc ( θ )}{\frac{c o s ( θ )}{s i n ( θ )} + csc ( θ )}

Utiliser l'identité trigonométrique de base:

csc (x) = \frac{1}{sin ( x )}

= \frac{\frac{c o s ( θ )}{s i n ( θ )} - \frac{1}{s i n ( θ )}}{\frac{c o s ( θ )}{s i n ( θ )} + \frac{1}{s i n ( θ )}}

Simplifier

\frac{\frac{c o s ( θ )}{s i n ( θ )} - \frac{1}{s i n ( θ )}}{\frac{c o s ( θ )}{s i n ( θ )} + \frac{1}{s i n ( θ )}} : \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

\frac{\frac{c o s ( θ )}{s i n ( θ )} - \frac{1}{s i n ( θ )}}{\frac{c o s ( θ )}{s i n ( θ )} + \frac{1}{s i n ( θ )}}

Combiner les fractions

\frac{cos ( θ )}{sin ( θ )} + \frac{1}{sin ( θ )} : \frac{cos ( θ ) + 1}{sin ( θ )}

Appliquer la règle

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

= \frac{cos ( θ ) + 1}{sin ( θ )}

= \frac{\frac{c o s ( θ )}{s i n ( θ )} - \frac{1}{s i n ( θ )}}{\frac{c o s ( θ ) + 1}{s i n ( θ )}}

Combiner les fractions

\frac{cos ( θ )}{sin ( θ )} - \frac{1}{sin ( θ )} : \frac{cos ( θ ) - 1}{sin ( θ )}

Appliquer la règle

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

= \frac{cos ( θ ) - 1}{sin ( θ )}

= \frac{\frac{c o s ( θ ) - 1}{s i n ( θ )}}{\frac{c o s ( θ ) + 1}{s i n ( θ )}}

Diviser des fractions:

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

= \frac{( cos ( θ ) - 1 ) sin ( θ )}{sin ( θ ) ( cos ( θ ) + 1 )}

Annuler le facteur commun :

sin (θ)

= \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

= \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

= \frac{- 1 + cos ( θ )}{1 + cos ( θ )}

En manipulant le côté droit

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

Récrire en utilisant des identités trigonométriques

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

Utiliser l'identité hyperbolique:

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

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

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

Simplifier

\frac{1 - 2 cos ( θ ) + cos ^{2} ( θ )}{- ( 1 - cos ^{2} ( θ ) )} : \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

\frac{1 - 2 cos ( θ ) + cos ^{2} ( θ )}{- ( 1 - cos ^{2} ( θ ) )}

Appliquer la règle des fractions:

\frac{a}{- b} = - \frac{a}{b}

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

Retirer les parenthèses:

(a) = a

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

Annuler

\frac{1 - 2 cos ( θ ) + cos ^{2} ( θ )}{1 - cos ^{2} ( θ )} : - \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

\frac{1 - 2 cos ( θ ) + cos ^{2} ( θ )}{1 - cos ^{2} ( θ )}

Factoriser

1 - 2 cos (θ) + cos^{2} (θ) : (cos (θ) - 1)^{2}

1 - 2 cos (θ) + cos^{2} (θ)

Ecrire sous la forme standard

a x^{2} + b x + c

= cos^{2} (θ) - 2 cos (θ) + 1

Récrire

cos^{2} (θ) - 2 cos (θ) + 1

comme

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

cos^{2} (θ) - 2 cos (θ) + 1

Récrire

1

comme

1^{2}

= cos^{2} (θ) - 2 cos (θ) + 1^{2}

Récrire

2 cos (θ)

comme

2 cos (θ) \cdot 1

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

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

Appliquer la formule du carré parfait:

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

a = cos (θ), b = 1

= (cos (θ) - 1)^{2}

= \frac{( cos ( θ ) - 1 ) ^{2}}{1 - cos ^{2} ( θ )}

Factoriser

1 - cos^{2} (θ) : - (cos (θ) + 1) (cos (θ) - 1)

1 - cos^{2} (θ)

Factoriser le terme commun

- 1

= - (cos^{2} (θ) - 1)

Factoriser

cos^{2} (θ) - 1 : (cos (θ) + 1) (cos (θ) - 1)

cos^{2} (θ) - 1

Récrire

1

comme

1^{2}

= cos^{2} (θ) - 1^{2}

Appliquer la formule de différence de deux carrés :

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

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

= (cos (θ) + 1) (cos (θ) - 1)

= - (cos (θ) + 1) (cos (θ) - 1)

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

Annuler le facteur commun :

cos (θ) - 1

= - \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

= - (- \frac{cos ( θ ) - 1}{cos ( θ ) + 1})

Appliquer la règle

- (- a) = a

= \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

= \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

= \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

Nous avons démontré que les deux côtés pourraient avoir la même forme

\Rightarrow vrai

Exemples populaires

prouver sin^2(θ/2)=(csc(θ)-cot(θ))/(2csc(θ))

p ro v e sin^{2} (\frac{θ}{2}) = \frac{csc ( θ ) - cot ( θ )}{2 csc ( θ )}

prouver cos(2x)=(cot^2(x)-1)/(cot^2(x)+1)

p ro v e cos (2 x) = \frac{cot ^{2} ( x ) - 1}{cot ^{2} ( x ) + 1}

prouver sin^2(x)-2cos^2(x)=3sin^2(x)-2

p ro v e sin^{2} (x) - 2 cos^{2} (x) = 3 sin^{2} (x) - 2

prouver tan(a)=tan(a)csc^2(a)+cot(-a)

p ro v e tan (a) = tan (a) csc^{2} (a) + cot (- a)

prouver 7csc^2(x)-5cot^2(x)=2csc^2(x)+5

p ro v e 7 csc^{2} (x) - 5 cot^{2} (x) = 2 csc^{2} (x) + 5