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受欢迎的三角函数 >

证明 sin(x)sec(x)= 1/(cos(x)csc(x))

解答

证明

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

解答

真

求解步骤

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

调整左侧

sin (x) sec (x)

用 sin, cos 表示

sec (x) sin (x)

使用基本三角恒等式:

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

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

化简

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

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

使用三角恒等式改写

使用基本三角恒等式:

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

\frac{\frac{1}{c s c ( x )}}{cos ( x )}

化简

\frac{\frac{1}{c s c ( x )}}{cos ( x )}

使用分式法则:

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

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

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

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

= \frac{1}{cos ( x ) csc ( 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 (1+tan^2(x))cos(x)=sec(x)

p ro v e (1 + tan^{2} (x)) cos (x) = sec (x)

证明 2/(cos^2(x))=2sec^2(x)

p ro v e \frac{2}{cos ^{2} ( x )} = 2 sec^{2} (x)

证明 2cos^2(a)=(1+sec(2a))cos(2a)

p ro v e 2 cos^{2} (a) = (1 + sec (2 a)) cos (2 a)

证明 sin((13pi)/2+x)=cos(x)

p ro v e sin (\frac{13 π}{2} + x) = cos (x)

证明 tan^2(-θ)+1=sec^2(θ)

p ro v e tan^{2} (- θ) + 1 = sec^{2} (θ)