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développer (1-cot^4(x))/(csc^2(x)(sin(x)+cos(x)))

Solution

développer

\frac{1 - cot ^{4} ( x )}{csc ^{2} ( x ) ( sin ( x ) + cos ( x ) )}

csc (x) (1 - cot (x))

étapes des solutions

\frac{1 - cot ^{4} ( x )}{csc ^{2} ( x ) ( sin ( x ) + cos ( x ) )}

Développer

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

= \frac{1 - cot ^{4} ( x )}{csc ^{2} ( x ) sin ( x ) + csc ^{2} ( x ) cos ( x )}

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

csc^{2} (x) cos (x) = cot (x) csc (x)

= \frac{1 - cot ^{4} ( x )}{csc ( x ) + cot ( x ) csc ( x )}

Factoriser

1 - cot^{4} (x) : - (cot^{2} (x) + 1) (cot (x) + 1) (cot (x) - 1)

= - \frac{( cot ^{2} ( x ) + 1 ) ( cot ( x ) + 1 ) ( cot ( x ) - 1 )}{csc ( x ) + cot ( x ) csc ( x )}

Factoriser le terme commun

csc (x)

= - \frac{( cot ^{2} ( x ) + 1 ) ( cot ( x ) + 1 ) ( cot ( x ) - 1 )}{csc ( x ) ( cot ( x ) + 1 )}

Annuler le facteur commun :

cot (x) + 1

= - \frac{( cot ^{2} ( x ) + 1 ) ( cot ( x ) - 1 )}{csc ( x )}

Utiliser l'identité trigonométrique de base:

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

= - (cot^{2} (x) + 1) (cot (x) - 1) sin (x)

Simplifier

cot^{2} (x) + 1 : csc^{2} (x)

= - csc^{2} (x) sin (x) (cot (x) - 1)

Appliquer la loi de la distribution:

a (b - c) = ab - a c

a = - csc^{2} (x) sin (x), b = cot (x), c = 1

= - csc^{2} (x) sin (x) cot (x) - (- csc^{2} (x) sin (x)) \cdot 1

Appliquer les règles des moins et des plus

- (- a) = a

= - csc^{2} (x) sin (x) cot (x) + 1 \cdot csc^{2} (x) sin (x)

Multiplier:

1 \cdot csc^{2} (x) = csc^{2} (x)

= - csc^{2} (x) sin (x) cot (x) + csc^{2} (x) sin (x)

Utiliser l'identité trigonométrique de base:

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

= csc^{2} (x) \frac{1}{csc ( x )}

Simplifier

csc^{2} (x) \frac{1}{csc ( x )} : csc (x)

= - csc^{2} (x) sin (x) cot (x) + csc (x)

csc (x) sin (x) = 1

= csc (x) (1 - cot (x) \cdot 1)

Factoriser

- csc^{2} (x) sin (x) cot (x) + csc (x) : csc (x) (- csc (x) sin (x) cot (x) + 1)

= csc (x) (- csc (x) sin (x) cot (x) + 1)

= csc (x) (1 - cot (x))

e x p an d \frac{x ^{2} + 2 x - 2}{( x - 1 ) ( x ^{2} - 4 x + 2 )}

e x p an d \frac{e ^{- 2 s}}{( s + 1 ) ( s + 2 )}

e x p an d x (e^{2 x} + 3 e^{x} = 54)

e x p an d \frac{( x - 2 ) ( - 4 x - 1 )}{2}

s im pl i f y δ (x^{'} - x)