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受欢迎的三角函数 >

证明 sin^2(x)+sin^2(x)cot^2(x)=1

解答

证明

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

解答

真

求解步骤

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

调整左侧

sin^{2} (x) + sin^{2} (x) cot^{2} (x)

用 sin, cos 表示

sin^{2} (x) + cot^{2} (x) sin^{2} (x)

使用基本三角恒等式:

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

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

化简

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

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

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

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

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

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

使用指数法则:

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

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

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

分式相乘:

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

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

约分:

sin^{2} (x)

= cos^{2} (x)

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

= 1

= 1

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(x)= 1/(cot(x))

p ro v e tan (x) = \frac{1}{cot ( x )}

证明 sin^2(θ/2)=(sec(θ)-1)/(2sec(θ))

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

证明 cot(t)csc(t)=(csc^2(t)-1)/(cos(t))

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

证明 1/(1+sin(t))=(sec(t)-tan(t))sec(t)

p ro v e \frac{1}{1 + sin ( t )} = (sec (t) - tan (t)) sec (t)

证明 sin(pi/3)=(sqrt(3))/2

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