tan(arctanX+arctanY)=(X+Y)/(1-XY)想了很久,没转过来......还有sin(2arctanX)=2X/(1+X^2)

Question

tan(arctanX+arctanY)=(X+Y)/(1-XY)
想了很久,没转过来......
还有 sin(2arctanX)=2X/(1+X^2)

Answer 1

❶证明：tan(arctanX+arctanY)=(X+Y)/(1-XY)
证明：tan(arctanx+arctany)=(tanarctanx+tanarctany)/[1-(tanarctanx)(tanarctany)]
=(x+y)/(1-xy)＝右边.
【如果看不明白,可令arctanx=α,arctany=β,那么tanα=tanarctanx=x,tanβ=tanarctany=y
❷证明sin(2arctanx)=2x/(1+x²)
证明：2sin(arctanx)cos(arctanx)=2(x/√(1+x²)][√(1/1+x²)]=2x/(1+x²)=右边.
这是因为α=arctanx,-π/2