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prove (sec(x))/(-csc(x))+(-sin(x))/(cos(x))=-2tan(x)

الحلّ

prove

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

صحيح

خطوات الحلّ

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

قم بالعمل على الطرف الأيسر

\frac{sec ( x )}{- csc ( x )} + \frac{- sin ( x )}{cos ( x )}

sin, cos :

عبّر بواسطة

\frac{- sin ( x )}{cos ( x )} + \frac{sec ( x )}{- csc ( x )}

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

:Use the basic trigonometric identity

= \frac{- sin ( x )}{cos ( x )} + \frac{\frac{1}{c o s ( x )}}{- csc ( x )}

csc (x) = \frac{1}{sin ( x )}

:Use the basic trigonometric identity

= \frac{- sin ( x )}{cos ( x )} + \frac{\frac{1}{c o s ( x )}}{- \frac{1}{s i n ( x )}}

\frac{- sin ( x )}{cos ( x )} + \frac{\frac{1}{c o s ( x )}}{- \frac{1}{s i n ( x )}}

بسّط:

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

\frac{- sin ( x )}{cos ( x )} + \frac{\frac{1}{c o s ( x )}}{- \frac{1}{s i n ( x )}}

\frac{- a}{b} = - \frac{a}{b}

: استخدم ميزات الكسور التالية

= - \frac{sin ( x )}{cos ( x )} + \frac{\frac{1}{c o s ( x )}}{- \frac{1}{s i n ( x )}}

\frac{\frac{1}{c o s ( x )}}{- \frac{1}{s i n ( x )}} = - \frac{sin ( x )}{cos ( x )}

\frac{\frac{1}{c o s ( x )}}{- \frac{1}{s i n ( x )}}

\frac{a}{- b} = - \frac{a}{b}

: استخدم ميزات الكسور التالية

= - \frac{\frac{1}{c o s ( x )}}{\frac{1}{s i n ( x )}}

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

:اقسم الكسور

= - \frac{1 \cdot sin ( x )}{cos ( x ) \cdot 1}

بسّط

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

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

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

فعّل القانون

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

- sin (x) - sin (x) = - 2 sin (x) :

اجمع العناصر المتشابهة

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

\frac{- a}{b} = - \frac{a}{b}

: استخدم ميزات الكسور التالية

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

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

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

Rewrite using trig identities

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

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

:Use the basic trigonometric identity

- 2 tan (x)

- 2 tan (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e 1 = \frac{sec ( u ) + tan ( u )}{sec ( u ) + tan ( u )}

p ro v e \frac{sin ^{2} ( x )}{1} \cdot \frac{1}{sin ^{2} ( x )} = 1

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

p ro v e cos (x + \frac{π}{6}) = \frac{3}{2} cos (x) - \frac{1}{2} sin (x)

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