Trigonometry Equation

1.
2cosθ - cotθ = 0
2cosθ = cotθ
2cosθ = cosθ/sinθ
2sinθcosθ - cosθ = 0
cosθ(2sinθ - 1) = 0
cosθ = 0 or sinθ = 0.5
θ = pi/2 or 3pi/2 or pi/6 or 5pi/6

2.
R.H.S
= (1 + sinθ)/(cosθ/sinθ - 1/sinθ) + (1 - sinθ)/(cosθ/sinθ + 1/sinθ)
= (1 + sinθ)*sinθ/(cosθ - 1) + (1 - sinθ)*sinθ/(cosθ + 1)
= sinθ((1 + sinθ)/(cosθ - 1) + (1 - sinθ)/(cosθ + 1))
= sinθ((1 + sinθ)(cosθ + 1) + (1 - sinθ)(cosθ - 1))/((cosθ + 1)(cosθ - 1))
= sinθ(1 + sinθ + cosθ + sinθcosθ - 1 + sinθ + cosθ - sinθcosθ)/(cos^2 θ - 1)
= sinθ(2sinθ + 2cosθ)/(-sin^2 θ)
= -2(sin^2 θ + sinθcosθ)/(sin^2 θ)
= -2(1 + cosθ/sinθ)
= -2(1 + cotθ)
= L.H.S

Remarks, cos^2 θ = cosθ * cosθ

2cosθ - cotθ = 0
2cosθ - cosθ/sinθ = 0
2cosθ = cosθ/sinθ
2sinθ=cosθ/cosθ
sinθ= 1/2
θ=60,180-60
so θ is 60 and 120

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L.H.S.=(1 + sinθ) / (cotθ - cscθ) + (1 - sinθ) / (cotθ + cscθ)
=(1+sinθ)(cotθ+cscθ) + (1-sinθ)(cotθ-cscθ) / (cotθ)^2 - (cscθ)^2
=cotθ+cscθ+sinθcotθ+sinθcscθ+cotθ-cscθ-sinθcotθ+sinθcscθ / cot^2 θ - csc^2 θ
=2cotθ+2sinθcscθ / cot^2 θ - csc^2 θ
=2cotθ+2 / cot^2 θ - csc^2 θ
=2cotθ+2 / cot^2 θ - (1 + cot^2 θ)
=2(1+cotθ) / (-1)
=-2(1 + cotθ)
=R.H.S

2007-04-09 16:42:04 補充：
For Q1,I type wrong. sinθ= 1/2θ=60,90＋60so θ is 60 and 150Dear first answerer,I think that pi/2 and 3pi/2 this two answers should be rejectedbecause cot pi/2 and cot 3pi/2 can not be counted.