Question

Question 1:-

Cos A - Sin A+ 1 / Cos A + Sin A- 1 = Cosec A + Cot A , Using identity cosec² A = 1 + cot² A

Note:- Solve in copy
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Answer 1

Step-by-step explanation:

★Given

• $\rm{ \dfrac{ cosA - sinA + 1}{cos A + sin A - 1} }$

★To prove -

• $\rm{Cosec A + Cot A}$

★Concept -

• Here, we'll use the trigonometric identity cosec² A = 1 + cot² A to solve the question. Let's do it!!

★Solution -

L.H.S

$\longmapsto\rm{ \dfrac{ cosA - sinA + 1}{cos A + sin A - 1} }$

$\longmapsto\rm{ \dfrac{ \dfrac{cosA - sinA + 1}{sin A } }{ \dfrac{cos A + sin A - 1}{sin A } } }$

$\longmapsto \rm{ \dfrac{cot A - 1 + cosec A }{cot A + 1 - cosec A } }$

$\boxed{ \longmapsto \rm{ \dfrac{cot A + cosec A - (cosec ^{2}A - cot ^{2} A) }{cot A + 1 - cosec A } }}$

∵ cosec²A = 1 + cot² A

=> cosec²A - cot²A = 1

${ \longmapsto \rm{ \dfrac{cot A + cosec A - [ (cosec A + cot A) (cosec A - cot A)] }{cot A + 1 - cosec A } }}$

${ \longmapsto \rm{ \dfrac{(cosec A + cot A) \cancel{ (1 - cosec A + cot A)] }} { \cancel{cot A + 1 - cosec A } }}}$

$\longmapsto \rm{ \bf \underline \orange{cosec A + cot A = R.H.S}}$