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受欢迎的三角函数 >

证明 (cot(y))/(sec(y))=csc(y)-sin(y)

解答

证明

\frac{cot ( y )}{sec ( y )} = csc (y) - sin (y)

解答

真

求解步骤

\frac{cot ( y )}{sec ( y )} = csc (y) - sin (y)

调整左侧

\frac{cot ( y )}{sec ( y )}

用 sin, cos 表示

\frac{cot ( y )}{sec ( y )}

使用基本三角恒等式:

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

= \frac{\frac{c o s ( y )}{s i n ( y )}}{sec ( y )}

使用基本三角恒等式:

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

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

化简

\frac{\frac{c o s ( y )}{s i n ( y )}}{\frac{1}{c o s ( y )}} : \frac{cos ^{2} ( y )}{sin ( y )}

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

分式相除:

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

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

整理后得

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

cos (y) cos (y) = cos^{2} (y)

cos (y) cos (y)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

cos (y) cos (y) = cos^{1 + 1} (y)

= cos^{1 + 1} (y)

数字相加:

1 + 1 = 2

= cos^{2} (y)

= \frac{cos ^{2} ( y )}{sin ( y )}

= \frac{cos ^{2} ( y )}{sin ( y )}

= \frac{cos ^{2} ( y )}{sin ( y )}

调整右侧

csc (y) - sin (y)

用 sin, cos 表示

csc (y) - sin (y)

使用基本三角恒等式:

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

= \frac{1}{sin ( y )} - sin (y)

化简

\frac{1}{sin ( y )} - sin (y) : \frac{1 - sin ^{2} ( y )}{sin ( y )}

\frac{1}{sin ( y )} - sin (y)

将项转换为分式:

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

= \frac{1}{sin ( y )} - \frac{sin ( y ) sin ( y )}{sin ( y )}

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 - sin ( y ) sin ( y )}{sin ( y )}

1 - sin (y) sin (y) = 1 - sin^{2} (y)

1 - sin (y) sin (y)

sin (y) sin (y) = sin^{2} (y)

sin (y) sin (y)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

sin (y) sin (y) = sin^{1 + 1} (y)

= sin^{1 + 1} (y)

数字相加:

1 + 1 = 2

= sin^{2} (y)

= 1 - sin^{2} (y)

= \frac{1 - sin ^{2} ( y )}{sin ( y )}

= \frac{1 - sin ^{2} ( y )}{sin ( y )}

= \frac{1 - sin ^{2} ( y )}{sin ( y )}

使用三角恒等式改写

\frac{1 - sin ^{2} ( y )}{sin ( y )}

使用毕达哥拉斯恒等式:

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

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

= \frac{cos ^{2} ( y )}{sin ( y )}

= \frac{cos ^{2} ( y )}{sin ( y )}

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 csc(θ)=cos(θ)cot(θ)+sin(θ)

p ro v e csc (θ) = cos (θ) cot (θ) + sin (θ)

证明 sin(z+pi/2)=cos(z)

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

证明 (sec(α))/(1-cos(α))=(sec(α)+1)/(sin^2(α))

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

证明 cos(pi/4)=(sqrt(2))/2

p ro v e cos (\frac{π}{4}) = \frac{2}{2}

证明 cot(x)=(csc(x))/(sec(x))

p ro v e cot (x) = \frac{csc ( x )}{sec ( x )}