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受欢迎的三角函数 >

证明 ((tan^2(t)))/(sec(t))=sec(t)-cos(t)

解答

证明

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

解答

真

求解步骤

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

调整左侧

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

用 sin, cos 表示

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

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

\frac{( \frac{s i n ( t )}{c o s ( t )} ) ^{2}}{\frac{1}{c o s ( t )}} : \frac{sin ^{2} ( t )}{cos ( t )}

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

使用分式法则:

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

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

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

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

使用指数法则:

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

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

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

乘

\frac{sin ^{2} ( t )}{cos ^{2} ( t )} cos (t) : \frac{sin ^{2} ( t )}{cos ( t )}

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

分式相乘:

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

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

约分:

cos (t)

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

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

使用分式法则:

\frac{a}{1} = a

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

= \frac{1 - cos ^{2} ( t )}{cos ( t )}

= \frac{1 - cos ^{2} ( t )}{cos ( t )}

调整右侧

sec (t) - cos (t)

用 sin, cos 表示

- cos (t) + sec (t)

使用基本三角恒等式:

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

= - cos (t) + \frac{1}{cos ( t )}

化简

- cos (t) + \frac{1}{cos ( t )} : \frac{- cos ^{2} ( t ) + 1}{cos ( t )}

- cos (t) + \frac{1}{cos ( t )}

将项转换为分式:

cos (t) = \frac{cos ( t ) cos ( t )}{cos ( t )}

= - \frac{cos ( t ) cos ( t )}{cos ( t )} + \frac{1}{cos ( t )}

因为分母相等，所以合并分式:

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

= \frac{- cos ( t ) cos ( t ) + 1}{cos ( t )}

- cos (t) cos (t) + 1 = - cos^{2} (t) + 1

- cos (t) cos (t) + 1

cos (t) cos (t) = cos^{2} (t)

cos (t) cos (t)

使用指数法则:

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

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

= cos^{1 + 1} (t)

数字相加:

1 + 1 = 2

= cos^{2} (t)

= - cos^{2} (t) + 1

= \frac{- cos ^{2} ( t ) + 1}{cos ( t )}

= \frac{- cos ^{2} ( t ) + 1}{cos ( t )}

= \frac{1 - cos ^{2} ( t )}{cos ( t )}

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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