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Question
Prove that: `1/(cosec"A" - cot"A") - 1/sin"A" = 1/sin"A" - 1/(cosec"A" + cot"A")`
Solution
= `1/("cosecA" - cot"A") - 1/sin"A"`
= `("cosec"^2"A" - cot^2"A")/("cosecA" - cot"A") - "cosecA"`
= `(("cosecA" - cot"A")("cosecA" + cot"A"))/("cosecA" - cot"A") - "cosecA"`
cosecA + cotA − cosecA
= cotA
R.H.S. = `1/sin"A" - 1/("cosecA" + cot"A")`
= `"cosecA" - (("cosec"^2"A" - cot"A")("cosecA" + cot"A"))/("cosecA" + cot"A")`
= cosecA − cosecA + cosecA
= cotA
= L.H.S.
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