Question

`(cot "A" + "cosec  A" - 1)/(cot"A" - "cosec  A" + 1) = (1 + cos "A")/"sin A"` हे सिद्ध करा.

Answer 1

डावी बाजू = `(cot "A" + "cosec A" - 1)/(cot"A" - "cosec A" + 1)`

= `(cot"A" + "cosec A" - ("cosec"^2"A" - cot^2"A"))/(cot"A" - "cosec A" + 1)`   .....`[(because 1 + cot^2"A" = "cosec"^2"A"),(therefore "cosec"^2"A" - cot^2"A" = 1)]`

= `(cot"A" + "cosec A" - ("cosec A" + cot"A")("cosec A" - cot"A"))/(cot"A" - "cosec A" + 1)`   .....[∵ a² – b² = (a + b) (a – b)]

= `((cot"A" + "cosec A")(1 - "cosec A" + cot "A"))/(cot"A" - "cosec A" + 1)`

= cot A + cosec A

= `"cos A"/"sin A" + 1/"sin A"`

= `(cos "A" + 1)/"sin A"`

= उजवी बाजू

∴ `(cot "A" + "cosec  A" - 1)/(cot"A" - "cosec  A" + 1) = (1 + cos "A")/"sin A"`