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العربية

prove coth^2(x)-1=csch^2(x)

الحلّ

prove

coth^{2} (x) - 1 = csch^{2} (x)

صحيح

خطوات الحلّ

coth^{2} (x) - 1 = csch^{2} (x)

قم بالعمل على الطرف الأيسر

coth^{2} (x) - 1

Rewrite using trig identities

coth^{2} (x) - 1

coth (x) = \frac{cosh ( x )}{sinh ( x )}

:Use the Hyperbolic identity

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

\frac{sinh ^{2} ( x )}{sinh ^{2} ( x )}

كـ

1

اكتب مجددًا

= (\frac{cosh ( x )}{sinh ( x )})^{2} - \frac{sinh ^{2} ( x )}{sinh ^{2} ( x )}

(\frac{cosh ( x )}{sinh ( x )})^{2} - \frac{sinh ^{2} ( x )}{sinh ^{2} ( x )}

بسّط:

\frac{cosh ^{2} ( x ) - sinh ^{2} ( x )}{sinh ^{2} ( x )}

(\frac{cosh ( x )}{sinh ( x )})^{2} - \frac{sinh ^{2} ( x )}{sinh ^{2} ( x )}

\frac{a}{a} = 1

فعّل القانون

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

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

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

:فعّل قانون القوى

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

اضرب

= \frac{cosh ^{2} ( x ) - sinh ^{2} ( x )}{sinh ^{2} ( x )}

= \frac{cosh ^{2} ( x ) - sinh ^{2} ( x )}{sinh ^{2} ( x )}

= \frac{cosh ^{2} ( x ) - sinh ^{2} ( x )}{sinh ^{2} ( x )}

Rewrite using trig identities

\frac{cosh ^{2} ( x ) - sinh ^{2} ( x )}{sinh ^{2} ( x )}

cosh^{2} (x) - sinh^{2} (x) = 1

:Use the Hyperbolic identity

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

sinh (x) = \frac{1}{csch ( x )}

:Use the Hyperbolic identity

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

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

بسّط:

csch^{2} (x)

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

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

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

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

:فعّل قانون القوى

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

1^{a} = 1

فعّل القانون

1^{2} = 1

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

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

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

: استخدم ميزات الكسور التالية

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

\frac{a}{1} = a

فعّل القانون

= csch^{2} (x)

= csch^{2} (x)

= csch^{2} (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e \frac{cos ^{4} ( x ) - sin ^{4} ( x )}{cos ^{2} ( x )} = 1 - tan^{2} (x)

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

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

p ro v e cot (a) sec (a) = csc (a)

p ro v e tan (\frac{x}{2}) + cot (\frac{x}{2}) = 2 csc (x)