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受欢迎的三角函数 >

证明 sin^2(x)+1/(sec^2(x))=sin(x)csc(x)

解答

证明

sin^{2} (x) + \frac{1}{sec ^{2} ( x )} = sin (x) csc (x)

解答

真

求解步骤

sin^{2} (x) + \frac{1}{sec ^{2} ( x )} = sin (x) csc (x)

调整左侧

sin^{2} (x) + \frac{1}{sec ^{2} ( x )}

用 sin, cos 表示

\frac{1}{sec ^{2} ( x )} + sin^{2} (x)

使用基本三角恒等式:

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

= \frac{1}{( \frac{1}{c o s ( x )} ) ^{2}} + sin^{2} (x)

化简

\frac{1}{( \frac{1}{c o s ( x )} ) ^{2}} + sin^{2} (x) : cos^{2} (x) + sin^{2} (x)

\frac{1}{( \frac{1}{c o s ( x )} ) ^{2}} + sin^{2} (x)

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

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

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

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

使用指数法则:

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

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

使用法则

1^{a} = 1

1^{2} = 1

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

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

使用分式法则:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

使用法则

\frac{a}{1} = a

= cos^{2} (x)

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

= 1

= 1

调整右侧

sin (x) csc (x)

用 sin, cos 表示

csc (x) sin (x)

使用基本三角恒等式:

csc (x) = \frac{1}{sin ( x )}

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

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

\frac{1}{sin ( x )} sin (x)

分式相乘:

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

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

约分:

sin (x)

= 1

= 1

= 1

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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