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受欢迎的三角函数 >

证明 5cos^2(x)-5sin^2(x)=10cos^2(x)-5

解答

证明

5 cos^{2} (x) - 5 sin^{2} (x) = 10 cos^{2} (x) - 5

解答

真

求解步骤

5 cos^{2} (x) - 5 sin^{2} (x) = 10 cos^{2} (x) - 5

调整左侧

5 cos^{2} (x) - 5 sin^{2} (x)

使用三角恒等式改写

5 cos^{2} (x) - 5 sin^{2} (x)

使用毕达哥拉斯恒等式:

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

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

= 5 cos^{2} (x) - 5 (1 - cos^{2} (x))

化简

5 cos^{2} (x) - 5 (1 - cos^{2} (x)) : 10 cos^{2} (x) - 5

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

乘开

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

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

使用分配律:

a (b - c) = ab - a c

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

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

使用加减运算法则

- (- a) = a

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

数字相乘:

5 \cdot 1 = 5

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

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

化简

5 cos^{2} (x) - 5 + 5 cos^{2} (x) : 10 cos^{2} (x) - 5

5 cos^{2} (x) - 5 + 5 cos^{2} (x)

对同类项分组

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

同类项相加:

5 cos^{2} (x) + 5 cos^{2} (x) = 10 cos^{2} (x)

= 10 cos^{2} (x) - 5

= 10 cos^{2} (x) - 5

= 10 cos^{2} (x) - 5

= 10 cos^{2} (x) - 5

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-cos(2x)))/((sin(2x)))

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

证明 cot(pi-θ)=-cot(θ)

p ro v e cot (π - θ) = - cot (θ)

证明 sin(x+2pi)=sin(x)

p ro v e sin (x + 2 π) = sin (x)

证明 (cos(3a)cos(2a))/(sin(3a)sin(2a))=(cot(2a))/(tan(3a))

p ro v e \frac{cos ( 3 a ) cos ( 2 a )}{sin ( 3 a ) sin ( 2 a )} = \frac{cot ( 2 a )}{tan ( 3 a )}

证明 1/(cos^2(θ))=tan^2(θ)+1

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