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受欢迎的三角函数 >

证明 (csc(x)+sec(x))/(tan(x)+1)=csc(x)

解答

证明

\frac{csc ( x ) + sec ( x )}{tan ( x ) + 1} = csc (x)

解答

真

求解步骤

\frac{csc ( x ) + sec ( x )}{tan ( x ) + 1} = csc (x)

调整左侧

\frac{csc ( x ) + sec ( x )}{tan ( x ) + 1}

用 sin, cos 表示

\frac{csc ( x ) + sec ( x )}{1 + tan ( x )}

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

化简

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

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

将项转换为分式:

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

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

乘以:

1 \cdot cos (x) = cos (x)

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

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

化简

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

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

sin (x), cos (x)

的最小公倍数:

sin (x) cos (x)

sin (x), cos (x)

最小公倍数 (LCM)

计算出由出现在

sin (x)

或

cos (x)

中的因子组成的表达式

= sin (x) cos (x)

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

sin (x) cos (x)

对于

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

将分母和分子乘以

cos (x)

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

对于

\frac{1}{cos ( x )} :

将分母和分子乘以

sin (x)

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

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

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

分式相除:

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

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

约分:

cos (x) + sin (x)

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

约分:

cos (x)

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

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

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

使用三角恒等式改写

使用基本三角恒等式:

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

\frac{1}{\frac{1}{c s c ( x )}}

化简

\frac{1}{\frac{1}{c s c ( x )}}

使用分式法则:

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

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

使用法则

\frac{a}{1} = a

= csc (x)

csc (x)

csc (x)

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 cos(pi/3)= 1/2

p ro v e cos (\frac{π}{3}) = \frac{1}{2}

证明 sec(-x)=sec(x)

p ro v e sec (- x) = sec (x)

证明 cos(pi/3+x)+cos(pi/3-x)=cos(x)

p ro v e cos (\frac{π}{3} + x) + cos (\frac{π}{3} - x) = cos (x)

证明 cos^2(2x)=(1+cos(4x))/2

p ro v e cos^{2} (2 x) = \frac{1 + cos ( 4 x )}{2}

证明 sin^2(θ)csc^2(θ)-sin^2(θ)=cos^2(θ)

p ro v e sin^{2} (θ) csc^{2} (θ) - sin^{2} (θ) = cos^{2} (θ)