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العربية

prove 4/(cos^2(x))-5=4tan^2(x)-1

الحلّ

prove

\frac{4}{cos ^{2} ( x )} - 5 = 4 tan^{2} (x) - 1

صحيح

خطوات الحلّ

\frac{4}{cos ^{2} ( x )} - 5 = 4 tan^{2} (x) - 1

قم بالعمل على الطرف الأيمن

4 tan^{2} (x) - 1

sin, cos :

عبّر بواسطة

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

4 (\frac{sin ( x )}{cos ( x )})^{2} = \frac{4 sin ^{2} ( x )}{cos ^{2} ( x )}

4 (\frac{sin ( x )}{cos ( x )})^{2}

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

(\frac{sin ( x )}{cos ( x )})^{2}

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

:فعّل قانون القوى

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

= 4 \cdot \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

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

:اضرب كسور

= \frac{sin ^{2} ( x ) \cdot 4}{cos ^{2} ( x )}

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

1 \cdot cos^{2} (x) = cos^{2} (x) :

اضرب

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

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

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

Rewrite using trig identities

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

cos^{2} (x) + sin^{2} (x) = 1

:فعّل نطريّة فيتاغوروس

sin^{2} (x) = 1 - cos^{2} (x)

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

- cos^{2} (x) + 4 (1 - cos^{2} (x))

وسٌع:

- 5 cos^{2} (x) + 4

- cos^{2} (x) + 4 (1 - cos^{2} (x))

4 (1 - cos^{2} (x))

وسٌع:

4 - 4 cos^{2} (x)

4 (1 - cos^{2} (x))

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = 4, b = 1, c = cos^{2} (x)

= 4 \cdot 1 - 4 cos^{2} (x)

4 \cdot 1 = 4 :

اضرب الأعداد

= 4 - 4 cos^{2} (x)

= - cos^{2} (x) + 4 - 4 cos^{2} (x)

- cos^{2} (x) + 4 - 4 cos^{2} (x)

بسّط:

- 5 cos^{2} (x) + 4

- cos^{2} (x) + 4 - 4 cos^{2} (x)

جمّع التعابير المتشابهة

= - cos^{2} (x) - 4 cos^{2} (x) + 4

- cos^{2} (x) - 4 cos^{2} (x) = - 5 cos^{2} (x) :

اجمع العناصر المتشابهة

= - 5 cos^{2} (x) + 4

= - 5 cos^{2} (x) + 4

= \frac{- 5 cos ^{2} ( x ) + 4}{cos ^{2} ( x )}

= \frac{- 5 cos ^{2} ( x ) + 4}{cos ^{2} ( x )}

\frac{4 - 5 cos ^{2} ( x )}{cos ^{2} ( x )}

وسٌع:

\frac{4}{cos ^{2} ( x )} - 5

\frac{4 - 5 cos ^{2} ( x )}{cos ^{2} ( x )}

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

: استخدم ميزات الكسور التالية

\frac{4 - 5 cos ^{2} ( x )}{cos ^{2} ( x )} = \frac{4}{cos ^{2} ( x )} - \frac{5 cos ^{2} ( x )}{cos ^{2} ( x )}

= \frac{4}{cos ^{2} ( x )} - \frac{5 cos ^{2} ( x )}{cos ^{2} ( x )}

\frac{5 cos ^{2} ( x )}{cos ^{2} ( x )}

اختزل:

5

\frac{5 cos ^{2} ( x )}{cos ^{2} ( x )}

cos^{2} (x) :

إلغ العوامل المشتركة

= 5

= \frac{4}{cos ^{2} ( x )} - 5

= - 5 + \frac{4}{cos ^{2} ( x )}

= \frac{4}{cos ^{2} ( x )} - 5

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

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

p ro v e 25 sin^{2} (x) + 25 cos^{2} (x) = 25

p ro v e cot (x) sin (x) sec (x) = 1

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

p ro v e tan^{2} (t) = \frac{1}{cos ^{2} ( t )} - 1