证明 cos(3x)+cos(x)=2cos(2x)cos(x)

解答

证明

cos (3 x) + cos (x) = 2 cos (2 x) cos (x)

真

求解步骤

cos (3 x) + cos (x) = 2 cos (2 x) cos (x)

调整左侧

cos (3 x) + cos (x)

使用三角恒等式改写

cos (3 x) + cos (x)

使用和差化积恒等式:

cos (s) + cos (t) = 2 cos (\frac{s + t}{2}) cos (\frac{s - t}{2})

= 2 cos (\frac{3 x + x}{2}) cos (\frac{3 x - x}{2})

化简

2 cos (\frac{3 x + x}{2}) cos (\frac{3 x - x}{2}) : 2 cos (2 x) cos (x)

2 cos (\frac{3 x + x}{2}) cos (\frac{3 x - x}{2})

\frac{3 x + x}{2} = 2 x

\frac{3 x + x}{2}

p ro v e sin (4 x) = 2 sin (2 x) cos (2 x)

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

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

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

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