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受欢迎的三角函数 >

证明 9sec^2(x)-5tan^2(x)=5+4sec^2(x)

解答

证明

9 sec^{2} (x) - 5 tan^{2} (x) = 5 + 4 sec^{2} (x)

解答

真

求解步骤

9 sec^{2} (x) - 5 tan^{2} (x) = 5 + 4 sec^{2} (x)

调整左侧

9 sec^{2} (x) - 5 tan^{2} (x)

使用三角恒等式改写

9 sec^{2} (x) - 5 tan^{2} (x)

使用毕达哥拉斯恒等式:

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

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

= 9 sec^{2} (x) - 5 (sec^{2} (x) - 1)

化简

9 sec^{2} (x) - 5 (sec^{2} (x) - 1) : 4 sec^{2} (x) + 5

9 sec^{2} (x) - 5 (sec^{2} (x) - 1)

乘开

- 5 (sec^{2} (x) - 1) : - 5 sec^{2} (x) + 5

- 5 (sec^{2} (x) - 1)

使用分配律:

a (b - c) = ab - a c

a = - 5, b = sec^{2} (x), c = 1

= - 5 sec^{2} (x) - (- 5) \cdot 1

使用加减运算法则

- (- a) = a

= - 5 sec^{2} (x) + 5 \cdot 1

数字相乘:

5 \cdot 1 = 5

= - 5 sec^{2} (x) + 5

= 9 sec^{2} (x) - 5 sec^{2} (x) + 5

同类项相加:

9 sec^{2} (x) - 5 sec^{2} (x) = 4 sec^{2} (x)

= 4 sec^{2} (x) + 5

= 4 sec^{2} (x) + 5

= 4 sec^{2} (x) + 5

= 5 + 4 sec^{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 sin(3A)=3sin(A)-4sin^3(A)

p ro v e sin (3 A) = 3 sin (A) - 4 sin^{3} (A)

证明 cot^4(x)+cot^2(x)=cot^2(x)csc^2(x)

p ro v e cot^{4} (x) + cot^{2} (x) = cot^{2} (x) csc^{2} (x)

证明 cot^3(x)=cot(x)(csc^2(x)-1)

p ro v e cot^{3} (x) = cot (x) (csc^{2} (x) - 1)

证明 sin(2a)=(2tan(a))/(1+tan^2(a))

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

证明 tan(x)=((1-cos(2x)))/(sin(2x))

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