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Question

Find x from the following equations:
(i) cosecπ2+θ+x cos θ cotπ2+θ=sinπ2+θ(ii) x cotπ2+θ+tanπ2+θsin θ+cosecπ2+θ=0

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Solution

90°=π2
i We have: cosec90°+θ + x cos θ cot90°+θ = sin90°+θ sec θ + x cos θ -tan θ = cos θ sec θ - x cosθ tanθ = cos θ sec θ - x cosθ×sin θcos θ = cos θ sec θ - x sin θ = cos θ sec θ - cos θ = x sin θ 1cos θ - cosθ = x sin θ 1 - cos2 θcos θ = x sin θ sin2θcosθ = x sin θ sin2 θcos θ sin θ = x sin θcos θ = x tan θ = x x =tan θ


ii We have: x cot90°+θ +tan90°+θ sin θ +cosec90°+θ =0 x -tan θ +-cot θ sin θ +sec θ =0 - x tan θ -cot θ sin θ +sec θ =0 - x×sin θcos θ -cos θ sin θ × sin θ +1 cos θ=0 - x×sin θcos θ -cos θ +1 cos θ =0 - x sin θ-cos2 θ+1cos θ =0
-x sin θ-cos2θ+1=0-xsin θ+sin2θ=0xsin θ=sin2θx=sin2 θsin θx=sin θ

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