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شائع الجبر >

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

الحلّ

expand

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

الحلّ

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)

خطوات الحلّ

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

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

وسّع:

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))

فعّل قانون ضرب الأقواس

= (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}

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

وسّع:

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}

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

وسّع:

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}

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

وسّع:

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}

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

وسّع:

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)

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)

بسّط:

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)

أمثلة شائعة

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