化简 (1+tan2θ)(1−sinθ)(1+sinθ)
待办事项
我们需要化简 (1+tan2θ)(1−sinθ)(1+sinθ).
解答
我们知道,
sin2θ+cos2θ=1.....(i)
sec2θ−tan2θ=1.......(ii)
secθ×cosθ=1.......(iii)
因此,
(1+tan2θ)(1−sinθ)(1+sinθ)=(1+tan2θ)(12−sin2θ) [因为 (a−b)(a+b)=a2−b2]
=(sec2θ)(cos2θ) [根据 (i) 和 (ii)]
=(secθ×cosθ)2
=12 [根据 (iii)]
=1
因此,
(1+tan2θ)(1−sinθ)(1+sinθ)=1.
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