English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

prove (sin(2x))/(sin(x))-(cos(2x))/(cos(x))=sec(x)

الحلّ

prove

\frac{sin ( 2 x )}{sin ( x )} - \frac{cos ( 2 x )}{cos ( x )} = sec (x)

صحيح

خطوات الحلّ

\frac{sin ( 2 x )}{sin ( x )} - \frac{cos ( 2 x )}{cos ( x )} = sec (x)

قم بالعمل على الطرف الأيسر

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

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

بسّط:

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

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

cos (x), sin (x)

المضاعف المشترك الأصغر لـ:

cos (x) sin (x)

cos (x), sin (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos (x)

sin (x)

= cos (x) sin (x)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

cos (x) sin (x)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{cos ( 2 x )}{cos ( x )} :

multiply the denominator and numerator by

sin (x)

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

For

\frac{sin ( 2 x )}{sin ( x )} :

multiply the denominator and numerator by

cos (x)

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

Rewrite using trig identities

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

sin (s) cos (t) - cos (s) sin (t) = sin (s - t)

:فعّل متطابقة الطرح لزوايا

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

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

بسّط:

\frac{1}{cos ( x )}

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

2 x - x = x :

اجمع العناصر المتشابهة

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

sin (x) :

إلغ العوامل المشتركة

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

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

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

Rewrite using trig identities

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

:Use the basic trigonometric identity

\frac{1}{\frac{1}{s e c ( x )}}

بسّط

\frac{1}{\frac{1}{s e c ( x )}}

\frac{1}{\frac{b}{c}} = \frac{c}{b}

: استخدم ميزات الكسور التالية

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

\frac{a}{1} = a

فعّل القانون

= sec (x)

sec (x)

sec (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e (csc^{2} (x) - 1) sec (x) = cos (x) csc^{2} (x)

p ro v e (1 + cot^{2} (x)) tan^{2} (x) = sec^{2} (x)

p ro v e sin (x) (csc (x) - sin (x)) = cos^{2} (x)

p ro v e cot (x) - tan (x) = sec (x) (csc (x) - 2 sin (x))

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