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受欢迎的三角函数 >

证明 (tan^2(θ))/(sec(θ))=sin(θ)tan(θ)

解答

证明

\frac{tan ^{2} ( θ )}{sec ( θ )} = sin (θ) tan (θ)

解答

真

求解步骤

\frac{tan ^{2} ( θ )}{sec ( θ )} = sin (θ) tan (θ)

调整左侧

\frac{tan ^{2} ( θ )}{sec ( θ )}

用 sin, cos 表示

\frac{tan ^{2} ( θ )}{sec ( θ )}

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

使用分式法则:

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

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

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

(\frac{sin ( θ )}{cos ( θ )})^{2}

使用指数法则:

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

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

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

乘

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

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

分式相乘:

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

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

约分:

cos (θ)

= \frac{sin ^{2} ( θ )}{cos ( θ )}

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

使用分式法则:

\frac{a}{1} = a

= \frac{sin ^{2} ( θ )}{cos ( θ )}

= \frac{sin ^{2} ( θ )}{cos ( θ )}

= \frac{sin ^{2} ( θ )}{cos ( θ )}

使用三角恒等式改写

= sin (θ) \frac{sin ( θ )}{cos ( θ )}

使用基本三角恒等式:

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

sin (θ) tan (θ)

sin (θ) tan (θ)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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