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受欢迎的三角函数 >

证明 ((tan(x)cot(x)))/(cos(x))=sec(x)

解答

证明

\frac{( tan ( x ) cot ( x ) )}{cos ( x )} = sec (x)

解答

真

求解步骤

\frac{tan ( x ) cot ( x )}{cos ( x )} = sec (x)

调整左侧

\frac{tan ( x ) cot ( x )}{cos ( x )}

用 sin, cos 表示

\frac{cot ( x ) tan ( x )}{cos ( x )}

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

\frac{\frac{c o s ( x )}{s i n ( x )} \cdot \frac{s i n ( x )}{c o s ( x )}}{cos ( x )} : \frac{1}{cos ( x )}

\frac{\frac{c o s ( x )}{s i n ( x )} \cdot \frac{s i n ( x )}{c o s ( x )}}{cos ( x )}

乘

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

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

分式相乘:

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

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

约分:

cos (x)

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

约分:

sin (x)

= 1

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

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

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

使用三角恒等式改写

使用基本三角恒等式:

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

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

化简

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

使用分式法则:

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

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

使用法则

\frac{a}{1} = a

= sec (x)

sec (x)

sec (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 cot(t/2)=(sin(t))/(1-cos(t))

p ro v e cot (\frac{t}{2}) = \frac{sin ( t )}{1 - cos ( t )}

证明 1-(cos^2(x))/(1+sin^2(x))=sin(x)

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

证明 1/2 sec(x)-1=0

p ro v e \frac{1}{2} sec (x) - 1 = 0

证明 (cos(2x)+1)/2 =cos^2(x)

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

证明 cos^3(A)= 1/4 (cos(3A)+3cos(A))

p ro v e cos^{3} (A) = \frac{1}{4} (cos (3 A) + 3 cos (A))