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受欢迎的三角函数 >

证明 1/(sin(x)+1)+1/(csc(x)+1)=1

解答

证明

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

解答

真

求解步骤

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

调整左侧

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

用 sin, cos 表示

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

使用基本三角恒等式:

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

= \frac{1}{1 + \frac{1}{s i n ( x )}} + \frac{1}{1 + sin ( x )}

\frac{1}{1 + \frac{1}{s i n ( x )}} + \frac{1}{1 + sin ( x )} = 1

\frac{1}{1 + \frac{1}{s i n ( x )}} + \frac{1}{1 + sin ( x )}

\frac{1}{1 + \frac{1}{s i n ( x )}} = \frac{sin ( x )}{sin ( x ) + 1}

\frac{1}{1 + \frac{1}{s i n ( x )}}

化简

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

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

将项转换为分式:

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

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

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{1 \cdot sin ( x ) + 1}{sin ( x )}

乘以:

1 \cdot sin (x) = sin (x)

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

= \frac{1}{\frac{s i n ( x ) + 1}{s i n ( x )}}

使用分式法则:

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

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

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

使用法则

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

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

使用法则

\frac{a}{a} = 1

= 1

= 1

= 1

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

流行的例子

证明 sin(t)csc(t)=1

p ro v e sin (t) csc (t) = 1

证明 cos(x)=sin(x+pi/2)

p ro v e cos (x) = sin (x + \frac{π}{2})

证明 sec(2θ)=(sec^2(θ))/(2-sec^2(θ))

p ro v e sec (2 θ) = \frac{sec ^{2} ( θ )}{2 - sec ^{2} ( θ )}

证明 tan(x)+cot(x)= 2/(sin(2x))

p ro v e tan (x) + cot (x) = \frac{2}{sin ( 2 x )}

证明 tan(2x)= 2/(cot(x)-tan(x))

p ro v e tan (2 x) = \frac{2}{cot ( x ) - tan ( x )}