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العربية

prove sin(x)cos(x)+sin^3(x)sec(x)=tan(x)

الحلّ

prove

sin (x) cos (x) + sin^{3} (x) sec (x) = tan (x)

صحيح

خطوات الحلّ

sin (x) cos (x) + sin^{3} (x) sec (x) = tan (x)

قم بالعمل على الطرف الأيسر

sin (x) cos (x) + sin^{3} (x) sec (x)

sin, cos :

عبّر بواسطة

cos (x) sin (x) + sec (x) sin^{3} (x)

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

:Use the basic trigonometric identity

= cos (x) sin (x) + \frac{1}{cos ( x )} sin^{3} (x)

cos (x) sin (x) + \frac{1}{cos ( x )} sin^{3} (x)

بسّط:

\frac{cos ^{2} ( x ) sin ( x ) + sin ^{3} ( x )}{cos ( x )}

cos (x) sin (x) + \frac{1}{cos ( x )} sin^{3} (x)

\frac{1}{cos ( x )} sin^{3} (x) = \frac{sin ^{3} ( x )}{cos ( x )}

\frac{1}{cos ( x )} sin^{3} (x)

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

:اضرب كسور

= \frac{1 \cdot sin ^{3} ( x )}{cos ( x )}

1 \cdot sin^{3} (x) = sin^{3} (x) :

اضرب

= \frac{sin ^{3} ( x )}{cos ( x )}

= cos (x) sin (x) + \frac{sin ^{3} ( x )}{cos ( x )}

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

:حوّل الأعداد لكسور

= \frac{cos ( x ) sin ( x ) cos ( x )}{cos ( x )} + \frac{sin ^{3} ( x )}{cos ( x )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{cos ( x ) sin ( x ) cos ( x ) + sin ^{3} ( x )}{cos ( x )}

cos (x) sin (x) cos (x) + sin^{3} (x) = cos^{2} (x) sin (x) + sin^{3} (x)

cos (x) sin (x) cos (x) + sin^{3} (x)

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

cos (x) sin (x) cos (x)

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

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

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

1 + 1 = 2 :

اجمع الأعداد

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

= cos^{2} (x) sin (x) + sin^{3} (x)

= \frac{cos ^{2} ( x ) sin ( x ) + sin ^{3} ( x )}{cos ( x )}

= \frac{cos ^{2} ( x ) sin ( x ) + sin ^{3} ( x )}{cos ( x )}

= \frac{sin ^{3} ( x ) + cos ^{2} ( x ) sin ( x )}{cos ( x )}

\frac{sin ^{3} ( x ) + cos ^{2} ( x ) sin ( x )}{cos ( x )}

حلل إلى عوامل:

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

\frac{sin ^{3} ( x ) + cos ^{2} ( x ) sin ( x )}{cos ( x )}

sin^{3} (x) + cos^{2} (x) sin (x)

حلل إلى عوامل:

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

sin^{3} (x) + cos^{2} (x) sin (x)

a^{b + c} = a^{b} a^{c}

:فعّل قانون القوى

sin^{3} (x) = sin (x) sin^{2} (x)

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

sin (x)

قم باخراج العامل المشترك

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

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

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

Rewrite using trig identities

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

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

:فعّل نطريّة فيتاغوروس

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

بسّط

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

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

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

:Use the basic trigonometric identity

= tan (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e - tan (θ) cos (θ) = sin (- θ)

p ro v e sec (t) - \frac{cos ( t )}{1 + sin ( t )} = tan (t)

p ro v e sin^{2} (x) csc^{2} (x) - sin^{2} (x) = cos^{2} (x)

p ro v e (sec (u) - tan (u)) (csc (u) + 1) = cot (u)

p ro v e sin (x) = cos (x - \frac{π}{2})