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شائع علم المثلثات >

tan(x)cot(x)=csc(x)sec(x)

الحلّ

tan (x) cot (x) = csc (x) sec (x)

الحلّ

x \in R لا يوجد حلّ لـ

خطوات الحلّ

tan (x) cot (x) = csc (x) sec (x)

من الطرفين

csc (x) sec (x)

اطرح

tan (x) cot (x) - csc (x) sec (x) = 0

sin, cos :

عبّر بواسطة

cot (x) tan (x) - csc (x) sec (x)

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

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

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

:اضرب كسور

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

cos (x) :

إلغ العوامل المشتركة

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

sin (x) :

إلغ العوامل المشتركة

= 1

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

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

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

:اضرب كسور

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

1 \cdot 1 = 1 :

اضرب الأعداد

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

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

اضرب

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

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

\frac{- 1 + cos ( x ) sin ( x )}{cos ( x ) sin ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- 1 + cos (x) sin (x) = 0

Rewrite using trig identities

- 1 + cos (x) sin (x)

2 sin (x) cos (x) = sin (2 x)

:فعّل متطابقة الزاوية المضاعفة

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

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

- 1 + \frac{sin ( 2 x )}{2} = 0

انقل

1

إلى الجانب الأيمن

- 1 + \frac{sin ( 2 x )}{2} = 0

للطرفين

1

أضف

- 1 + \frac{sin ( 2 x )}{2} + 1 = 0 + 1

بسّط

\frac{sin ( 2 x )}{2} = 1

\frac{sin ( 2 x )}{2} = 1

2

اضرب الطرفين بـ

\frac{sin ( 2 x )}{2} = 1

2

اضرب الطرفين بـ

\frac{2 sin ( 2 x )}{2} = 1 \cdot 2

بسّط

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

x \in R لا يوجد حلّ لـ

رسم

أمثلة شائعة

cos(A)= 3/5

cos (A) = \frac{3}{5}

cos^2(x)-sin^2(x)= 1/(sqrt(2))

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

arctan(2x)+arctan(x)= pi/2

arctan (2 x) + arctan (x) = \frac{π}{2}

2cos(x)-5=3cos(x)-4

2 cos (x) - 5 = 3 cos (x) - 4

pi/2 =arctan(x)

\frac{π}{2} = arctan (x)