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العربية

prove 1-cos^2(x)=(tan^2(x))/(sec^2(x))

الحلّ

prove

صحيح

خطوات الحلّ

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

قم بالعمل على الطرف الأيمن

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

sin, cos :

عبّر بواسطة

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

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

:Use the basic trigonometric identity

= \frac{( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{sec ^{2} ( x )}

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

:Use the basic trigonometric identity

= \frac{( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{( \frac{1}{c o s ( x )} ) ^{2}}

\frac{( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{( \frac{1}{c o s ( x )} ) ^{2}}

بسّط:

sin^{2} (x)

\frac{( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{( \frac{1}{c o s ( x )} ) ^{2}}

(\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{( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{\frac{1}{c o s ^{2} ( x )}}

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

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

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

:فعّل قانون القوى

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

= \frac{\frac{s i n ^{2} ( x )}{c o s ^{2} ( x )}}{\frac{1}{c o s ^{2} ( x )}}

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

:اقسم الكسور

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

بسّط

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

cos^{2} (x) :

إلغ العوامل المشتركة

= sin^{2} (x)

= sin^{2} (x)

= sin^{2} (x)

Rewrite using trig identities

sin^{2} (x)

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

:فعّل نطريّة فيتاغوروس

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

= 1 - cos^{2} (x)

= 1 - cos^{2} (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح