证明 1-2sin^2(x)+sin^4(x)=cos^4(x)

解答

证明

1 - 2 sin^{2} (x) + sin^{4} (x) = cos^{4} (x)

真

求解步骤

1 - 2 sin^{2} (x) + sin^{4} (x) = cos^{4} (x)

调整左侧

1 - 2 sin^{2} (x) + sin^{4} (x)

使用完全平方公式:

a^{2} - 2 ab + b^{2} = (a - b)^{2}

1 - 2 sin^{2} (x) + sin^{4} (x) = (1 - sin^{2} (x))^{2}

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

使用毕达哥拉斯恒等式:

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

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

= (cos^{2} (x))^{2}

化简

(cos^{2} (x))^{2} : cos^{4} (x)

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

使用指数法则:

(a^{b})^{c} = a^{b c}

= cos^{2 \cdot 2} (x)

数字相乘:

2 \cdot 2 = 4

= cos^{4} (x)

= cos^{4} (x)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

p ro v e \frac{sin ( 3 θ )}{sin ( θ )} = 3 - 4 sin^{2} (θ)

p ro v e tan (π - θ) = tan (θ)

p ro v e 6 cos^{2} (x) - 3 = 3 - 6 sin^{2} (x)

p ro v e sin (x) tan (x) + cos (- x) = sec (x)

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