证明 (cos(θ)+cos(3θ))/(2cos(θ))=cos(2θ)

解答

证明

\frac{cos ( θ ) + cos ( 3 θ )}{2 cos ( θ )} = cos (2 θ)

真

求解步骤

\frac{cos ( θ ) + cos ( 3 θ )}{2 cos ( θ )} = cos (2 θ)

调整左侧

\frac{cos ( θ ) + cos ( 3 θ )}{2 cos ( θ )}

使用三角恒等式改写

\frac{cos ( θ ) + cos ( 3 θ )}{2 cos ( θ )}

使用和差化积恒等式:

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

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

化简

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

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

2 cos (\frac{θ + 3 θ}{2}) cos (\frac{θ - 3 θ}{2}) = 2 cos (\frac{4 θ}{2}) cos (- \frac{2 θ}{2})

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

p ro v e 3 tan (x) = 4 sin (x)

p ro v e sin (x) + cos (x) = 2 sin (x + \frac{π}{4})

p ro v e (sin (t)) cot (t) = cos (t)

p ro v e cos (θ) = 1 - sin^{2} (θ)

p ro v e \frac{cot ^{3} ( t )}{csc ( t )} = cos (t) cot^{2} (t)