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受欢迎的三角函数 >

证明 sin^2(x)-2cos^2(x)=3sin^2(x)-2

解答

证明

sin^{2} (x) - 2 cos^{2} (x) = 3 sin^{2} (x) - 2

解答

真

求解步骤

sin^{2} (x) - 2 cos^{2} (x) = 3 sin^{2} (x) - 2

调整左侧

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

化简

sin^{2} (x) - 2 (1 - sin^{2} (x)) : 3 sin^{2} (x) - 2

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

乘开

- 2 (1 - sin^{2} (x)) : - 2 + 2 sin^{2} (x)

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

使用分配律:

a (b - c) = ab - a c

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

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

使用加减运算法则

- (- a) = a

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

数字相乘:

2 \cdot 1 = 2

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

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

化简

sin^{2} (x) - 2 + 2 sin^{2} (x) : 3 sin^{2} (x) - 2

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

对同类项分组

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

同类项相加:

sin^{2} (x) + 2 sin^{2} (x) = 3 sin^{2} (x)

= 3 sin^{2} (x) - 2

= 3 sin^{2} (x) - 2

= 3 sin^{2} (x) - 2

= 3 sin^{2} (x) - 2

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(a)=tan(a)csc^2(a)+cot(-a)

p ro v e tan (a) = tan (a) csc^{2} (a) + cot (- a)

证明 7csc^2(x)-5cot^2(x)=2csc^2(x)+5

p ro v e 7 csc^{2} (x) - 5 cot^{2} (x) = 2 csc^{2} (x) + 5

证明 1/(csc(x)+cot(x))=csc(x)-cot(x)

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

证明 tan(θ)(cot(θ)+tan(θ))=sec^2(θ)

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

证明 sec^2(-θ)-1=tan^2(θ)

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