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شائع الجبر >

expand (1-cot^4(x))/(csc^2(x)(sin(x)+cos(x)))

الحلّ

expand

\frac{1 - cot ^{4} ( x )}{csc ^{2} ( x ) ( sin ( x ) + cos ( x ) )}

الحلّ

csc (x) (1 - cot (x))

خطوات الحلّ

\frac{1 - cot ^{4} ( x )}{csc ^{2} ( x ) ( sin ( x ) + cos ( x ) )}

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

وسٌع:

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

= \frac{1 - cot ^{4} ( x )}{csc ^{2} ( x ) sin ( x ) + csc ^{2} ( x ) cos ( x )}

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

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

= \frac{1 - cot ^{4} ( x )}{csc ( x ) + cot ( x ) csc ( x )}

1 - cot^{4} (x)

حلل إلى عوامل:

- (cot^{2} (x) + 1) (cot (x) + 1) (cot (x) - 1)

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

csc (x)

قم باخراج العامل المشترك

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

cot (x) + 1 :

إلغ العوامل المشتركة

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

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

:Use the basic trigonometric identity

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

cot^{2} (x) + 1

بسّط:

csc^{2} (x)

= - csc^{2} (x) sin (x) (cot (x) - 1)

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = - csc^{2} (x) sin (x), b = cot (x), c = 1

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

فعّل قوانين سالب-موجب

- (- a) = a

= - csc^{2} (x) sin (x) cot (x) + 1 \cdot csc^{2} (x) sin (x)

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

اضرب

= - csc^{2} (x) sin (x) cot (x) + csc^{2} (x) sin (x)

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

:Use the basic trigonometric identity

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

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

بسّط:

csc (x)

= - csc^{2} (x) sin (x) cot (x) + csc (x)

csc (x) sin (x) = 1

= csc (x) (1 - cot (x) \cdot 1)

- csc^{2} (x) sin (x) cot (x) + csc (x)

حلل إلى عوامل:

csc (x) (- csc (x) sin (x) cot (x) + 1)

= csc (x) (- csc (x) sin (x) cot (x) + 1)

= csc (x) (1 - cot (x))

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