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prouver sin^2(a)+sin^2(a)tan^2(a)=tan^2(a)

Solution

prouver

sin^{2} (a) + sin^{2} (a) tan^{2} (a) = tan^{2} (a)

vrai

étapes des solutions

sin^{2} (a) + sin^{2} (a) tan^{2} (a) = tan^{2} (a)

En manipulant le côté gauche

sin^{2} (a) + sin^{2} (a) tan^{2} (a)

Récrire en utilisant des identités trigonométriques

sin^{2} (a) + sin^{2} (a) tan^{2} (a)

Utiliser l'identité hyperbolique:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

= sin^{2} (a) + sin^{2} (a) (sec^{2} (a) - 1)

Simplifier

sin^{2} (a) + sin^{2} (a) (sec^{2} (a) - 1) : sin^{2} (a) sec^{2} (a)

sin^{2} (a) + sin^{2} (a) (sec^{2} (a) - 1)

Développer

sin^{2} (a) (sec^{2} (a) - 1) : sin^{2} (a) sec^{2} (a) - sin^{2} (a)

sin^{2} (a) (sec^{2} (a) - 1)

Appliquer la loi de la distribution:

a (b - c) = ab - a c

a = sin^{2} (a), b = sec^{2} (a), c = 1

= sin^{2} (a) sec^{2} (a) - sin^{2} (a) \cdot 1

= sin^{2} (a) sec^{2} (a) - 1 \cdot sin^{2} (a)

Multiplier:

1 \cdot sin^{2} (a) = sin^{2} (a)

= sin^{2} (a) sec^{2} (a) - sin^{2} (a)

= sin^{2} (a) + sin^{2} (a) sec^{2} (a) - sin^{2} (a)

Additionner les éléments similaires :

sin^{2} (a) - sin^{2} (a) = 0

= sin^{2} (a) sec^{2} (a)

= sin^{2} (a) sec^{2} (a)

= sin^{2} (a) sec^{2} (a)

Exprimer avec sinus, cosinus

sec^{2} (a) sin^{2} (a)

Utiliser l'identité trigonométrique de base:

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

= (\frac{1}{cos ( a )})^{2} sin^{2} (a)

Simplifier

(\frac{1}{cos ( a )})^{2} sin^{2} (a) : \frac{sin ^{2} ( a )}{cos ^{2} ( a )}

(\frac{1}{cos ( a )})^{2} sin^{2} (a)

(\frac{1}{cos ( a )})^{2} = \frac{1}{cos ^{2} ( a )}

(\frac{1}{cos ( a )})^{2}

Appliquer la règle de l'exposant:

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

= \frac{1 ^{2}}{cos ^{2} ( a )}

Appliquer la règle

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( a )}

= \frac{1}{cos ^{2} ( a )} sin^{2} (a)

Multiplier des fractions:

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

= \frac{1 \cdot sin ^{2} ( a )}{cos ^{2} ( a )}

Multiplier:

1 \cdot sin^{2} (a) = sin^{2} (a)

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

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

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

Récrire en utilisant des identités trigonométriques

= \frac{sin ( a )}{cos ( a )} \cdot \frac{sin ( a )}{cos ( a )}

Utiliser l'identité trigonométrique de base:

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

\frac{sin ( a ) tan ( a )}{cos ( a )}

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

Utiliser l'identité trigonométrique de base:

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

tan (a) tan (a)

Simplifier

tan (a) tan (a) : tan^{2} (a)

tan (a) tan (a)

Appliquer la règle de l'exposant:

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

tan (a) tan (a) = tan^{1 + 1} (a)

= tan^{1 + 1} (a)

Additionner les nombres :

1 + 1 = 2

= tan^{2} (a)

tan^{2} (a)

tan^{2} (a)

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

\Rightarrow vrai

p ro v e sin (b) + cos (b) cot (b) = csc (b)

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

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

p ro v e tan (\frac{3 π}{2} - x) = cot (x)

p ro v e \frac{3 csc ( - x )}{sec ( - x )} = - 3 cot (x)