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شائع >

prove-csc^2(x)cos^2(x)=1-csc^2(x)

الحلّ

prove

- csc^{2} (x) cos^{2} (x) = 1 - csc^{2} (x)

صحيح

خطوات الحلّ

- csc^{2} (x) cos^{2} (x) = 1 - csc^{2} (x)

قم بالعمل على الطرف الأيمن

1 - csc^{2} (x)

sin, cos :

عبّر بواسطة

1 - csc^{2} (x)

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

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

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

:فعّل قانون القوى

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

1^{a} = 1

فعّل القانون

1^{2} = 1

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

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

1 \cdot sin^{2} (x) = sin^{2} (x) :

اضرب

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

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

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

Rewrite using trig identities

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

1 = cos^{2} (x) + sin^{2} (x)

:فعّل نطريّة فيتاغوروس

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

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

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

: استخدم ميزات الكسور التالية

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

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

Rewrite using trig identities

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

:Use the basic trigonometric identity

- \frac{cos ^{2} ( x )}{( \frac{1}{c s c ( x )} ) ^{2}}

بسّط

- \frac{cos ^{2} ( x )}{( \frac{1}{c s c ( x )} ) ^{2}}

(\frac{1}{csc ( x )})^{2} = \frac{1}{csc ^{2} ( x )}

(\frac{1}{csc ( x )})^{2}

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

:فعّل قانون القوى

= \frac{1 ^{2}}{csc ^{2} ( x )}

1^{a} = 1

فعّل القانون

1^{2} = 1

= \frac{1}{csc ^{2} ( x )}

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

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

: استخدم ميزات الكسور التالية

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

\frac{a}{1} = a

فعّل القانون

= - cos^{2} (x) csc^{2} (x)

- cos^{2} (x) csc^{2} (x)

- cos^{2} (x) csc^{2} (x)

= - csc^{2} (x) cos^{2} (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

sec (θ) = sin (θ) (tan (θ) + cot (θ))

\frac{csc ^{2} ( A )}{1 + tan ^{2} ( A )} = cot^{2} (A)

sin (3 a) = 3 sin (a) - 4 sin^{3} (a)

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

\frac{1}{tan ( θ ) csc ( θ )} = cos (θ)