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受欢迎的三角函数 >

证明 sec(t)-(cos(t))/(1+sin(t))=tan(t)

解答

证明

sec (t) - \frac{cos ( t )}{1 + sin ( t )} = tan (t)

解答

真

求解步骤

sec (t) - \frac{cos ( t )}{1 + sin ( t )} = tan (t)

调整左侧

sec (t) - \frac{cos ( t )}{1 + sin ( t )}

用 sin, cos 表示

- \frac{cos ( t )}{1 + sin ( t )} + sec (t)

使用基本三角恒等式:

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

= - \frac{cos ( t )}{1 + sin ( t )} + \frac{1}{cos ( t )}

化简

- \frac{cos ( t )}{1 + sin ( t )} + \frac{1}{cos ( t )} : \frac{- cos ^{2} ( t ) + sin ( t ) + 1}{cos ( t ) ( sin ( t ) + 1 )}

- \frac{cos ( t )}{1 + sin ( t )} + \frac{1}{cos ( t )}

1 + sin (t), cos (t)

的最小公倍数:

cos (t) (sin (t) + 1)

1 + sin (t), cos (t)

最小公倍数 (LCM)

计算出由出现在

1 + sin (t)

或

cos (t)

中的因子组成的表达式

= cos (t) (sin (t) + 1)

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

cos (t) (sin (t) + 1)

对于

\frac{cos ( t )}{1 + sin ( t )} :

将分母和分子乘以

cos (t)

\frac{cos ( t )}{1 + sin ( t )} = \frac{cos ( t ) cos ( t )}{( 1 + sin ( t ) ) cos ( t )} = \frac{cos ^{2} ( t )}{cos ( t ) ( sin ( t ) + 1 )}

对于

\frac{1}{cos ( t )} :

将分母和分子乘以

sin (t) + 1

\frac{1}{cos ( t )} = \frac{1 \cdot ( sin ( t ) + 1 )}{cos ( t ) ( sin ( t ) + 1 )} = \frac{sin ( t ) + 1}{cos ( t ) ( sin ( t ) + 1 )}

= - \frac{cos ^{2} ( t )}{cos ( t ) ( sin ( t ) + 1 )} + \frac{sin ( t ) + 1}{cos ( t ) ( sin ( t ) + 1 )}

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} ( t ) + sin ( t ) + 1}{cos ( t ) ( sin ( t ) + 1 )}

= \frac{- cos ^{2} ( t ) + sin ( t ) + 1}{cos ( t ) ( sin ( t ) + 1 )}

= \frac{1 - cos ^{2} ( t ) + sin ( t )}{( 1 + sin ( t ) ) cos ( t )}

使用三角恒等式改写

\frac{1 - cos ^{2} ( t ) + sin ( t )}{( 1 + sin ( t ) ) cos ( t )}

使用毕达哥拉斯恒等式:

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

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

= \frac{sin ^{2} ( t ) + sin ( t )}{( 1 + sin ( t ) ) cos ( t )}

化简

\frac{sin ^{2} ( t ) + sin ( t )}{( 1 + sin ( t ) ) cos ( t )} : \frac{sin ( t )}{cos ( t )}

\frac{sin ^{2} ( t ) + sin ( t )}{( 1 + sin ( t ) ) cos ( t )}

分解

sin^{2} (t) + sin (t) : sin (t) (sin (t) + 1)

sin^{2} (t) + sin (t)

使用指数法则:

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

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

= sin (t) sin (t) + sin (t)

因式分解出通项

sin (t)

= sin (t) (sin (t) + 1)

= \frac{sin ( t ) ( sin ( t ) + 1 )}{( 1 + sin ( t ) ) cos ( t )}

约分:

sin (t) + 1

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

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

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

使用基本三角恒等式:

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

= tan (t)

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 sin^2(x)csc^2(x)-sin^2(x)=cos^2(x)

p ro v e sin^{2} (x) csc^{2} (x) - sin^{2} (x) = cos^{2} (x)

证明 (sec(u)-tan(u))(csc(u)+1)=cot(u)

p ro v e (sec (u) - tan (u)) (csc (u) + 1) = cot (u)

证明 sin(x)=cos(x-pi/2)

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

证明 1/(1+cos(x))=csc^2(x)-csc(x)cot(x)

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

证明 1/(1-sin(x))-1/(1+sin(x))=(2tan(x))/(cos(x))

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