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العربية

prove tan^4(x)-sec^4(x)=-2sec^2(x)+1

الحلّ

prove

صحيح

خطوات الحلّ

tan^{4} (x) - sec^{4} (x) = - 2 sec^{2} (x) + 1

قم بالعمل على الطرف الأيسر

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

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

حلل إلى عوامل:

(tan^{2} (x) + sec^{2} (x)) (tan (x) + sec (x)) (tan (x) - sec (x))

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

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

كـ

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

اكتب مجددًا

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

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

:فعّل قانون القوى

sec^{4} (x) = (sec^{2} (x))^{2}

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

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

:فعّل قانون القوى

tan^{4} (x) = (tan^{2} (x))^{2}

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

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

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

فعّل قانون فرق المربّعات

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

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

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

حلل إلى عوامل:

(tan (x) + sec (x)) (tan (x) - sec (x))

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

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

فعّل قانون فرق المربّعات

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

= (tan (x) + sec (x)) (tan (x) - sec (x))

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

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

Rewrite using trig identities

(- sec (x) + tan (x)) (sec (x) + tan (x)) (sec^{2} (x) + tan^{2} (x))

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

:فعّل نطريّة فيتاغوروس

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

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

sec^{2} (x) + sec^{2} (x) = 2 sec^{2} (x) :

اجمع العناصر المتشابهة

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

(tan (x) + sec (x)) (tan (x) - sec (x))

وسٌع:

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

(tan (x) + sec (x)) (tan (x) - sec (x))

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

فعّل قانون فرق المربّعات

a = tan (x), b = sec (x)

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

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

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

:فعّل نطريّة فيتاغوروس

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

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

(- 1 + 2 sec^{2} (x)) (- 1)

بسّط:

1 - 2 sec^{2} (x)

(- 1 + 2 sec^{2} (x)) (- 1)

(- a) = - a

:احذف الأقواس

= - (- 1 + 2 sec^{2} (x)) \cdot 1

(- 1 + 2 sec^{2} (x)) \cdot 1 = (- 1 + 2 sec^{2} (x)) :

اضرب

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

افتح أقواس

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

فعّل قوانين سالب-موجب

- (- a) = a, - (a) = - a

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

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

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

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

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح