已知θ∈（0,π/2),且满足关系logtanθCOSθ=3/2 求logcsc^2θ(sinθ*cosθ)的值.

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已知θ∈（0,π/2),且满足关系logtanθCOSθ=3/2 求logcsc^2θ(sinθ*cosθ)的值.
答案是-7/10,要详细过程
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logtanθCOSθ
=1/logCOSθ tanθ
=1/logCOSθ (sinθ/cosθ)
=1/[logCOSθ (sinθ)-logCOSθ (cosθ)]
=1/[logCOSθ (sinθ)-1]
=3/2,
∴logCOSθ (sinθ)=5/3;
∴logsinθ (COSθ)=3/5;
则logcsc^2θ(sinθ*cosθ)
=(1/2)logcscθ(sinθ*cosθ)
=(-1/2)logsinθ(sinθ*cosθ)
=(-1/2)[1+logsinθ (cosθ)]
=(-1/2)[1+3/5]
=-4/5

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