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受欢迎的三角函数 >

证明 (cos(2X))/(sin(X))+(sin(2X))/(cos(X))=csc(X)

解答

证明

\frac{cos ( 2 X )}{sin ( X )} + \frac{sin ( 2 X )}{cos ( X )} = csc (X)

解答

真

求解步骤

\frac{cos ( 2 X )}{sin ( X )} + \frac{sin ( 2 X )}{cos ( X )} = csc (X)

调整左侧

\frac{cos ( 2 X )}{sin ( X )} + \frac{sin ( 2 X )}{cos ( X )}

使用三角恒等式改写

\frac{cos ( 2 X )}{sin ( X )} + \frac{sin ( 2 X )}{cos ( X )}

使用倍角公式:

sin (2 x) = 2 sin (x) cos (x)

= \frac{cos ( 2 X )}{sin ( X )} + \frac{2 sin ( X ) cos ( X )}{cos ( X )}

消掉

\frac{2 sin ( X ) cos ( X )}{cos ( X )} : 2 sin (X)

\frac{2 sin ( X ) cos ( X )}{cos ( X )}

约分:

cos (X)

= 2 sin (X)

= \frac{cos ( 2 X )}{sin ( X )} + 2 sin (X)

= \frac{cos ( 2 X )}{sin ( X )} + 2 sin (X)

调整右侧

csc (X)

用 sin, cos 表示

csc (X)

使用基本三角恒等式:

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

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

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

使用三角恒等式改写

\frac{1}{sin ( X )}

使用倍角公式:

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

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

= \frac{cos ( 2 X ) + 2 sin ^{2} ( X )}{sin ( X )}

= \frac{cos ( 2 X ) + 2 sin ^{2} ( X )}{sin ( X )}

乘开

\frac{cos ( 2 X ) + 2 sin ^{2} ( X )}{sin ( X )} : \frac{cos ( 2 X )}{sin ( X )} + 2 sin (X)

\frac{cos ( 2 X ) + 2 sin ^{2} ( X )}{sin ( X )}

使用分式法则:

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

\frac{cos ( 2 X ) + 2 sin ^{2} ( X )}{sin ( X )} = \frac{cos ( 2 X )}{sin ( X )} + \frac{2 sin ^{2} ( X )}{sin ( X )}

= \frac{cos ( 2 X )}{sin ( X )} + \frac{2 sin ^{2} ( X )}{sin ( X )}

消掉

\frac{2 sin ^{2} ( X )}{sin ( X )} : 2 sin (X)

\frac{2 sin ^{2} ( X )}{sin ( X )}

约分:

sin (X)

= 2 sin (X)

= \frac{cos ( 2 X )}{sin ( X )} + 2 sin (X)

= \frac{cos ( 2 X )}{sin ( X )} + 2 sin (X)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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