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شائع علم المثلثات >

prove csc^2(x)sec^2(x)=sec^2(x)+csc^2(x)

الحلّ

prove

csc^{2} (x) sec^{2} (x) = sec^{2} (x) + csc^{2} (x)

الحلّ

صحيح

خطوات الحلّ

csc^{2} (x) sec^{2} (x) = sec^{2} (x) + csc^{2} (x)

قم بالعمل على الطرف الأيمن

sec^{2} (x) + csc^{2} (x)

sin, cos :

عبّر بواسطة

csc^{2} (x) + sec^{2} (x)

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

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

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

:فعّل قانون القوى

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

1^{a} = 1

فعّل القانون

1^{2} = 1

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

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

sin^{2} (x), cos^{2} (x)

المضاعف المشترك الأصغر لـ:

sin^{2} (x) cos^{2} (x)

sin^{2} (x), cos^{2} (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin^{2} (x)

cos^{2} (x)

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

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sin^{2} (x) cos^{2} (x)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

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

multiply the denominator and numerator by

cos^{2} (x)

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

For

\frac{1}{cos ^{2} ( x )} :

multiply the denominator and numerator by

sin^{2} (x)

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

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

Rewrite using trig identities

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

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

:فعّل نطريّة فيتاغوروس

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

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

Rewrite using trig identities

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

بسّط

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

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

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

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

:فعّل قانون القوى

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

1^{a} = 1

فعّل القانون

1^{2} = 1

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

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

(\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{1}{\frac{1}{s e c ^{2} ( x )} \cdot \frac{1}{c s c ^{2} ( x )}}

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

اضرب بـ:

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

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

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

:اضرب كسور

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

1 \cdot 1 = 1 :

اضرب الأعداد

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

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

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

: استخدم ميزات الكسور التالية

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

\frac{a}{1} = a

فعّل القانون

= sec^{2} (x) csc^{2} (x)

sec^{2} (x) csc^{2} (x)

sec^{2} (x) csc^{2} (x)

= csc^{2} (x) sec^{2} (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

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