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עברית

العربية

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

الحلّ

prove

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

صحيح

خطوات الحلّ

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

قم بالعمل على الطرف الأيمن

\frac{tan ^{2} ( x )}{sec ( x ) - 1}

Rewrite using trig identities

\frac{tan ^{2} ( x )}{sec ( x ) - 1}

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

:فعّل نطريّة فيتاغوروس

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

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

\frac{sec ^{2} ( x ) - 1}{sec ( x ) - 1}

بسّط:

sec (x) + 1

\frac{sec ^{2} ( x ) - 1}{sec ( x ) - 1}

sec^{2} (x) - 1

حلل إلى عوامل:

(sec (x) + 1) (sec (x) - 1)

sec^{2} (x) - 1

1^{2}

كـ

1

اكتب مجددًا

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

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

فعّل قانون فرق المربّعات

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

= (sec (x) + 1) (sec (x) - 1)

= \frac{( sec ( x ) + 1 ) ( sec ( x ) - 1 )}{sec ( x ) - 1}

sec (x) - 1 :

إلغ العوامل المشتركة

= sec (x) + 1

= sec (x) + 1

= sec (x) + 1

sin, cos :

عبّر بواسطة

1 + sec (x)

sec (x) = \frac{1}{cos ( x )}

:Use the basic trigonometric identity

= 1 + \frac{1}{cos ( x )}

1 + \frac{1}{cos ( x )}

بسّط:

\frac{cos ( x ) + 1}{cos ( x )}

1 + \frac{1}{cos ( x )}

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

:حوّل الأعداد لكسور

= \frac{1 \cdot cos ( x )}{cos ( x )} + \frac{1}{cos ( x )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{1 \cdot cos ( x ) + 1}{cos ( x )}

1 \cdot cos (x) = cos (x) :

اضرب

= \frac{cos ( x ) + 1}{cos ( x )}

= \frac{cos ( x ) + 1}{cos ( x )}

= \frac{1 + cos ( x )}{cos ( x )}

\frac{1 + cos ( x )}{cos ( x )}

وسٌع:

\frac{1}{cos ( x )} + 1

\frac{1 + cos ( x )}{cos ( x )}

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

: استخدم ميزات الكسور التالية

\frac{1 + cos ( x )}{cos ( x )} = \frac{1}{cos ( x )} + \frac{cos ( x )}{cos ( x )}

= \frac{1}{cos ( x )} + \frac{cos ( x )}{cos ( x )}

\frac{a}{a} = 1

فعّل القانون

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

= \frac{1}{cos ( x )} + 1

= 1 + \frac{1}{cos ( x )}

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e sec (x) = sec (- x)

p ro v e cot^{2} (θ) = cos^{2} (θ) + (cot (θ) cos (θ))^{2}

p ro v e csc^{2} (θ) - 1 = cot^{2} (θ)

p ro v e tan (u) + cot (u) = sec (u) csc (u)

p ro v e cos (\frac{π}{12}) = \frac{2 + 3}{2}