1
Question
1+cos thita - sin thita/ cos thita -1+ sin thita = cosec thita + cot thita
1+cos thita - sin thita/ cos thita -1+ sin thita = cosec thita + cot thita
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Solution
LHS =
cosθ-sinθ+1 /cos θ + sinθ-1
Divide numerator and denominator by sinθ
=> cosecθ + Cotθ - 1 / cotθ - Cosecθ + 1
=> cosecθ + Cotθ - ( cosecθ^2 - Cotθ^2) / cotθ - Cosecθ + 1
=>cosecθ+ Cotθ - (cosecθ + Cotθ) (cosecθ - Cotθ) /cotθ - Cosecθ + 1
=>(cosecθ + Cotθ)( cotθ - Cosecθ + 1 ) / cotθ - cosecθ +1
=>cosecθ + Cotθ
LHS =
cosθ-sinθ+1 /cos θ + sinθ-1
Divide numerator and denominator by sinθ
=> cosecθ + Cotθ - 1 / cotθ - Cosecθ + 1
=> cosecθ + Cotθ - ( cosecθ^2 - Cotθ^2) / cotθ - Cosecθ + 1
=>cosecθ+ Cotθ - (cosecθ + Cotθ) (cosecθ - Cotθ) /cotθ - Cosecθ + 1
=>(cosecθ + Cotθ)( cotθ - Cosecθ + 1 ) / cotθ - cosecθ +1
=>cosecθ + Cotθ
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