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العربية

prove tan^2(x)=sin^2(x)+sin^2(x)tan^2(x)

الحلّ

prove

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

صحيح

خطوات الحلّ

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

قم بالعمل على الطرف الأيمن

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

Rewrite using trig identities

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

tan^{2} (x) + 1 = sec^{2} (x)

:فعّل نطريّة فيتاغوروس

tan^{2} (x) = sec^{2} (x) - 1

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

sin^{2} (x) + sin^{2} (x) (sec^{2} (x) - 1)

بسّط:

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

sin^{2} (x) + sin^{2} (x) (sec^{2} (x) - 1)

sin^{2} (x) (sec^{2} (x) - 1)

وسٌع:

sin^{2} (x) sec^{2} (x) - sin^{2} (x)

sin^{2} (x) (sec^{2} (x) - 1)

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = sin^{2} (x), b = sec^{2} (x), c = 1

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

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

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

اضرب

= sin^{2} (x) sec^{2} (x) - sin^{2} (x)

= sin^{2} (x) + sin^{2} (x) sec^{2} (x) - sin^{2} (x)

sin^{2} (x) - sin^{2} (x) = 0 :

اجمع العناصر المتشابهة

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

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

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

sin, cos :

عبّر بواسطة

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

:اضرب كسور

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

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

اضرب

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

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

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

Rewrite using trig identities

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

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

:Use the basic trigonometric identity

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

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

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

:Use the basic trigonometric identity

tan (x) tan (x)

tan (x) tan (x)

بسّط:

tan^{2} (x)

tan (x) tan (x)

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

:فعّل قانون القوى

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

= tan^{1 + 1} (x)

1 + 1 = 2 :

اجمع الأعداد

= tan^{2} (x)

tan^{2} (x)

tan^{2} (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

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p ro v e (sec^{2} (x) cot (x)) + cot (x) = tan (x)

p ro v e 6 cos^{2} (A) - 5 = 1 - 6 sin^{2} (A)