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受欢迎的三角函数 >

证明 sin^2(x)=(sec(x)sin(x))/(tan(x)+cot(x))

解答

证明

sin^{2} (x) = \frac{sec ( x ) sin ( x )}{tan ( x ) + cot ( x )}

解答

真

求解步骤

sin^{2} (x) = \frac{sec ( x ) sin ( x )}{tan ( x ) + cot ( x )}

调整右侧

\frac{sec ( x ) sin ( x )}{tan ( x ) + cot ( x )}

用 sin, cos 表示

\frac{sec ( x ) sin ( x )}{cot ( x ) + tan ( x )}

使用基本三角恒等式:

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

= \frac{\frac{1}{c o s ( x )} sin ( x )}{cot ( x ) + tan ( x )}

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

化简

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

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

将分母和分子乘以

cos (x)

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

对于

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

将分母和分子乘以

sin (x)

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

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

= \frac{\frac{1}{c o s ( x )} sin ( x )}{\frac{c o s ^{2} ( x ) + s i n ^{2} ( x )}{s i n ( x ) c o s ( x )}}

乘

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

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

分式相乘:

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

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

乘以:

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

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

= \frac{\frac{s i n ( x )}{c o s ( x )}}{\frac{c o s ^{2} ( x ) + s i n ^{2} ( 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{sin ( x ) sin ( x ) cos ( x )}{cos ( x ) ( cos ^{2} ( x ) + sin ^{2} ( x ) )}

约分:

cos (x)

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

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

sin (x) sin (x)

使用指数法则:

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

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

= sin^{1 + 1} (x)

数字相加:

1 + 1 = 2

= sin^{2} (x)

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

= \frac{sin ^{2} ( x )}{1}

使用法则

\frac{a}{1} = a

= sin^{2} (x)

= sin^{2} (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(A)=(cos(A))/(sin(A))

p ro v e cot (A) = \frac{cos ( A )}{sin ( A )}

证明 1-2cos^2(4)=2sin^2(4)-1

p ro v e 1 - 2 cos^{2} (4) = 2 sin^{2} (4) - 1

证明 sin(8a)=4sin(2a)cos(2a)cos(4a)

p ro v e sin (8 a) = 4 sin (2 a) cos (2 a) cos (4 a)

证明 csc^2(x)(tan(x))-(tan(x))=cot(x)

p ro v e csc^{2} (x) (tan (x)) - (tan (x)) = cot (x)

证明 (sin(3a))/(sin(a))=2cos(2a)+1

p ro v e \frac{sin ( 3 a )}{sin ( a )} = 2 cos (2 a) + 1