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受欢迎的三角函数 >

证明 (csc^2(θ))/(1+tan^2(θ))=cot^2(θ)

解答

证明

\frac{csc ^{2} ( θ )}{1 + tan ^{2} ( θ )} = cot^{2} (θ)

解答

真

求解步骤

\frac{csc ^{2} ( θ )}{1 + tan ^{2} ( θ )} = cot^{2} (θ)

调整左侧

\frac{csc ^{2} ( θ )}{1 + tan ^{2} ( θ )}

使用三角恒等式改写

\frac{csc ^{2} ( θ )}{1 + tan ^{2} ( θ )}

使用毕达哥拉斯恒等式:

tan^{2} (x) + 1 = sec^{2} (x)

= \frac{csc ^{2} ( θ )}{sec ^{2} ( θ )}

= \frac{csc ^{2} ( θ )}{sec ^{2} ( θ )}

用 sin, cos 表示

\frac{csc ^{2} ( θ )}{sec ^{2} ( θ )}

使用基本三角恒等式:

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

= \frac{( \frac{1}{s i n ( θ )} ) ^{2}}{sec ^{2} ( θ )}

使用基本三角恒等式:

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

= \frac{( \frac{1}{s i n ( θ )} ) ^{2}}{( \frac{1}{c o s ( θ )} ) ^{2}}

化简

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

\frac{( \frac{1}{s i n ( θ )} ) ^{2}}{( \frac{1}{c o s ( θ )} ) ^{2}}

(\frac{1}{cos ( θ )})^{2} = \frac{1}{cos ^{2} ( θ )}

(\frac{1}{cos ( θ )})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{cos ^{2} ( θ )}

使用法则

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( θ )}

= \frac{( \frac{1}{s i n ( θ )} ) ^{2}}{\frac{1}{c o s ^{2} ( θ )}}

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

(\frac{1}{sin ( θ )})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{sin ^{2} ( θ )}

使用法则

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( θ )}

= \frac{\frac{1}{s i n ^{2} ( θ )}}{\frac{1}{c o s ^{2} ( θ )}}

分式相除:

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

= \frac{1 \cdot cos ^{2} ( θ )}{sin ^{2} ( θ ) \cdot 1}

整理后得

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

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

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

使用三角恒等式改写

= \frac{cos ( θ )}{sin ( θ )} \cdot \frac{cos ( θ )}{sin ( θ )}

使用基本三角恒等式:

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

\frac{cos ( θ ) cot ( θ )}{sin ( θ )}

= cot (θ) \frac{cos ( θ )}{sin ( θ )}

使用基本三角恒等式:

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

cot (θ) cot (θ)

化简

cot (θ) cot (θ) : cot^{2} (θ)

cot (θ) cot (θ)

使用指数法则:

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

cot (θ) cot (θ) = cot^{1 + 1} (θ)

= cot^{1 + 1} (θ)

数字相加:

1 + 1 = 2

= cot^{2} (θ)

cot^{2} (θ)

cot^{2} (θ)

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 (tan(y))/(csc(y))=sec(y)-cos(y)

p ro v e \frac{tan ( y )}{csc ( y )} = sec (y) - cos (y)

证明 cos(3x)=cos(2x+x)

p ro v e cos (3 x) = cos (2 x + x)

证明 sin(8x)=2sin(4x)cos(4x)

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

证明 sin(t)csc(pi/2-t)=tan(t)

p ro v e sin (t) csc (\frac{π}{2} - t) = tan (t)

证明 (tan(x))/(csc(x))=sec(x)-cos(x)

p ro v e \frac{tan ( x )}{csc ( x )} = sec (x) - cos (x)