prove (csc(-x))/(sec(-x))=-cot(x)

الحلّ

prove

\frac{csc ( - x )}{sec ( - x )} = - cot (x)

صحيح

خطوات الحلّ

\frac{csc ( - x )}{sec ( - x )} = - cot (x)

قم بالعمل على الطرف الأيسر

\frac{csc ( - x )}{sec ( - x )}

csc (- x) = - csc (x)

:Use the negative angle identity

= \frac{- csc ( x )}{sec ( - x )}

sec (- x) = sec (x)

:Use the negative angle identity

= \frac{- csc ( x )}{sec ( x )}

sin, cos :

عبّر بواسطة

\frac{- csc ( x )}{sec ( x )}

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

: استخدم ميزات الكسور التالية

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

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

:اقسم الكسور

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

بسّط

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

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

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

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

:Use the basic trigonometric identity

= - cot (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e cos (x - y) + cos (x + y) = 2 cos (x) cos (y)

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

p ro v e (cot (x) - csc (x)) (cos (x) + 1) = - sin (x)

p ro v e csc^{4} (θ) - cot^{4} (θ) = 2 csc^{2} (θ) - 1

p ro v e \frac{cos ( θ ) cot ( θ )}{1 - sin ( θ )} - 1 = csc (θ)