expandir (sec(x)+tan(x))^3(sec(x)-tan(x))^4

Solução

expandir

(sec (x) + tan (x))^{3} (sec (x) - tan (x))^{4}

sec^{7} (x) + tan^{7} (x) - 11 sec^{3} (x) tan^{4} (x) - 11 tan^{3} (x) sec^{4} (x) - sec^{6} (x) tan (x) - 6 sec^{5} (x) tan^{2} (x) + 14 sec^{4} (x) tan^{3} (x) + 3 sec^{2} (x) tan^{5} (x) + 3 tan^{2} (x) sec^{5} (x) + 14 tan^{4} (x) sec^{3} (x) - 6 tan^{5} (x) sec^{2} (x) - tan^{6} (x) sec (x)

Passos da solução

(sec (x) + tan (x))^{3} (sec (x) - tan (x))^{4}

Expandir

(sec (x) + tan (x))^{3} : sec^{3} (x) + 3 sec^{2} (x) tan (x) + 3 tan^{2} (x) sec (x) + tan^{3} (x)

= (sec^{3} (x) + 3 sec^{2} (x) tan (x) + 3 tan^{2} (x) sec (x) + tan^{3} (x)) (sec (x) - tan (x))^{4}

= (sec (x) - tan (x))^{4} (sec^{3} (x) + 3 sec^{2} (x) tan (x) + 3 tan^{2} (x) sec (x) + tan^{3} (x))

Aplicar a seguinte regra dos produtos notáveis

= (sec (x) - tan (x))^{4} sec^{3} (x) + (sec (x) - tan (x))^{4} \cdot 3 sec^{2} (x) tan (x) + (sec (x) - tan (x))^{4} \cdot 3 tan^{2} (x) sec (x) + (sec (x) - tan (x))^{4} tan^{3} (x)

= sec^{3} (x) (sec (x) - tan (x))^{4} + 3 sec^{2} (x) tan (x) (sec (x) - tan (x))^{4} + 3 tan^{2} (x) sec (x) (sec (x) - tan (x))^{4} + tan^{3} (x) (sec (x) - tan (x))^{4}

Expandir

sec^{3} (x) (sec (x) - tan (x))^{4} : sec^{7} (x) - 4 sec^{6} (x) tan (x) + 6 sec^{5} (x) tan^{2} (x) - 4 sec^{4} (x) tan^{3} (x) + sec^{3} (x) tan^{4} (x)

= sec^{7} (x) - 4 sec^{6} (x) tan (x) + 6 sec^{5} (x) tan^{2} (x) - 4 sec^{4} (x) tan^{3} (x) + sec^{3} (x) tan^{4} (x) + 3 sec^{2} (x) tan (x) (sec (x) - tan (x))^{4} + 3 tan^{2} (x) sec (x) (sec (x) - tan (x))^{4} + tan^{3} (x) (sec (x) - tan (x))^{4}

Expandir

3 sec^{2} (x) tan (x) (sec (x) - tan (x))^{4} : 3 sec^{6} (x) tan (x) - 12 sec^{5} (x) tan^{2} (x) + 18 sec^{4} (x) tan^{3} (x) - 12 sec^{3} (x) tan^{4} (x) + 3 sec^{2} (x) tan^{5} (x)

= sec^{7} (x) - 4 sec^{6} (x) tan (x) + 6 sec^{5} (x) tan^{2} (x) - 4 sec^{4} (x) tan^{3} (x) + sec^{3} (x) tan^{4} (x) + 3 sec^{6} (x) tan (x) - 12 sec^{5} (x) tan^{2} (x) + 18 sec^{4} (x) tan^{3} (x) - 12 sec^{3} (x) tan^{4} (x) + 3 sec^{2} (x) tan^{5} (x) + 3 tan^{2} (x) sec (x) (sec (x) - tan (x))^{4} + tan^{3} (x) (sec (x) - tan (x))^{4}

Expandir

3 tan^{2} (x) sec (x) (sec (x) - tan (x))^{4} : 3 tan^{2} (x) sec^{5} (x) - 12 tan^{3} (x) sec^{4} (x) + 18 tan^{4} (x) sec^{3} (x) - 12 tan^{5} (x) sec^{2} (x) + 3 tan^{6} (x) sec (x)

= sec^{7} (x) - 4 sec^{6} (x) tan (x) + 6 sec^{5} (x) tan^{2} (x) - 4 sec^{4} (x) tan^{3} (x) + sec^{3} (x) tan^{4} (x) + 3 sec^{6} (x) tan (x) - 12 sec^{5} (x) tan^{2} (x) + 18 sec^{4} (x) tan^{3} (x) - 12 sec^{3} (x) tan^{4} (x) + 3 sec^{2} (x) tan^{5} (x) + 3 tan^{2} (x) sec^{5} (x) - 12 tan^{3} (x) sec^{4} (x) + 18 tan^{4} (x) sec^{3} (x) - 12 tan^{5} (x) sec^{2} (x) + 3 tan^{6} (x) sec (x) + tan^{3} (x) (sec (x) - tan (x))^{4}

Expandir

tan^{3} (x) (sec (x) - tan (x))^{4} : tan^{3} (x) sec^{4} (x) - 4 tan^{4} (x) sec^{3} (x) + 6 tan^{5} (x) sec^{2} (x) - 4 tan^{6} (x) sec (x) + tan^{7} (x)

= sec^{7} (x) - 4 sec^{6} (x) tan (x) + 6 sec^{5} (x) tan^{2} (x) - 4 sec^{4} (x) tan^{3} (x) + sec^{3} (x) tan^{4} (x) + 3 sec^{6} (x) tan (x) - 12 sec^{5} (x) tan^{2} (x) + 18 sec^{4} (x) tan^{3} (x) - 12 sec^{3} (x) tan^{4} (x) + 3 sec^{2} (x) tan^{5} (x) + 3 tan^{2} (x) sec^{5} (x) - 12 tan^{3} (x) sec^{4} (x) + 18 tan^{4} (x) sec^{3} (x) - 12 tan^{5} (x) sec^{2} (x) + 3 tan^{6} (x) sec (x) + tan^{3} (x) sec^{4} (x) - 4 tan^{4} (x) sec^{3} (x) + 6 tan^{5} (x) sec^{2} (x) - 4 tan^{6} (x) sec (x) + tan^{7} (x)

Simplificar

sec^{7} (x) - 4 sec^{6} (x) tan (x) + 6 sec^{5} (x) tan^{2} (x) - 4 sec^{4} (x) tan^{3} (x) + sec^{3} (x) tan^{4} (x) + 3 sec^{6} (x) tan (x) - 12 sec^{5} (x) tan^{2} (x) + 18 sec^{4} (x) tan^{3} (x) - 12 sec^{3} (x) tan^{4} (x) + 3 sec^{2} (x) tan^{5} (x) + 3 tan^{2} (x) sec^{5} (x) - 12 tan^{3} (x) sec^{4} (x) + 18 tan^{4} (x) sec^{3} (x) - 12 tan^{5} (x) sec^{2} (x) + 3 tan^{6} (x) sec (x) + tan^{3} (x) sec^{4} (x) - 4 tan^{4} (x) sec^{3} (x) + 6 tan^{5} (x) sec^{2} (x) - 4 tan^{6} (x) sec (x) + tan^{7} (x) : sec^{7} (x) + tan^{7} (x) - 11 sec^{3} (x) tan^{4} (x) - 11 tan^{3} (x) sec^{4} (x) - sec^{6} (x) tan (x) - 6 sec^{5} (x) tan^{2} (x) + 14 sec^{4} (x) tan^{3} (x) + 3 sec^{2} (x) tan^{5} (x) + 3 tan^{2} (x) sec^{5} (x) + 14 tan^{4} (x) sec^{3} (x) - 6 tan^{5} (x) sec^{2} (x) - tan^{6} (x) sec (x)

= sec^{7} (x) + tan^{7} (x) - 11 sec^{3} (x) tan^{4} (x) - 11 tan^{3} (x) sec^{4} (x) - sec^{6} (x) tan (x) - 6 sec^{5} (x) tan^{2} (x) + 14 sec^{4} (x) tan^{3} (x) + 3 sec^{2} (x) tan^{5} (x) + 3 tan^{2} (x) sec^{5} (x) + 14 tan^{4} (x) sec^{3} (x) - 6 tan^{5} (x) sec^{2} (x) - tan^{6} (x) sec (x)

s im pl i f y ((cos (θ) + i sin (θ)))^{3}

e x p an d c H (r (o, s, x), o, s, x) a d r (o, s, x) a d os (x - 2 x = 6)

s im pl i f y \frac{12.9}{( s + 1.1 ) ( s + 2 )}

s im pl i f y x^{3} (\frac{1}{x} - sin (\frac{1}{x}))

e x p an d \frac{x}{( x + 2 ) ( x + 3 )}