Question

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



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Answer 1

EXPLANATION.

$\sf \: \dfrac{ \cot(a) + \csc(a) - 1}{ \cot(a) - \csc(a) + 1} = \dfrac{1 + \cos(a) }{ \sin(a) } \\ \\ \sf \: \frac{ \cot(a) + \csc(a) - ( \csc {}^{2} (a) - \cot {}^{2} (a) )}{ \cot(a) - \csc(a) + 1} \\ \\ \sf \: \dfrac{ \cot(a) + \csc(a) - ( \csc(a) + \cot(a))( \csc(a) - \cot(a)) }{ \cot(a) - \csc(a) + 1} \\ \\ \sf \: \frac{ \cot(a) + \csc(a) (1 - \csc(a) + \cot(a) )}{ \cot(a) - \csc(a) + 1}$

$\sf \: \cot(a) + \csc(a) \\ \\ \sf \: \frac{ \cos(a) }{ \sin(a) } + \frac{1}{ \sin(a) } \\ \\ \sf \: \frac{1 + \cos(a) }{ \sin(a) } = proved$

More information.

= sin²A + cos²A = 1

= Tan²A + 1 = sec²A

= 1 + cot²A = csc²A

= sin(2A) = 2sinA.cosA

= cos(2A) = cos²A - sin²A

Answer 2

$\bigstar \underline{\underline{\sf \bf To \ Prove:-}}$

$\\: \implies \sf \frac{ (cot A + cosec A - 1)}{( cot A - cosec A +1 )} = \frac{( 1 + cos A)} {sin A}$

$\\\bigstar \underline{\underline{ \sf \bf Proof:-}}$

$\\ \sf LHS: \\\implies \sf \frac{ (cot A + cosec A - 1)}{( cot A - cosec A +1 )}$

We know :

❥ Cot A = CosA/SinA

❥ Cosec A = 1/SinA

Now ,

$\\\\ \implies \sf \frac{CosA/sinA + (1/sinA)-1}{CosA/sinA-(1/sinA)+1}$

$\implies \sf \frac{CosA+1-sinA / {SinA}}{CosA - 1+SinA/SinA} \\\\\\\implies \sf \frac{CosA+ 1-SinA}{CosA-1+SinA} \\\\\\\implies \sf \frac{(CosA-SinA)+1}{(CosA+SinA)-1}$

Rationalise it by (Cos A + Sin A) +1.

$\\\\ \implies \sf \frac{(CosA-SinA)+1}{(CosA+SinA)-1} \times \frac{(CosA+sinA)+1}{(CosA+sinA)+1}$

$\\ \implies \sf \frac{cos2A-(1- cos2A) + 2cosA+1}{(cos2A+sin2A+2sinA cosA) -1}\\\\\\ \implies \sf \frac{ cos2A-1+ cos2A + 2cosA+1}{2sinA cosA +1 -1}\\\\\\ \implies \sf \frac{2cos2A + 2cosA}{2sinA cosA}\\\\\\ \implies \sf \frac{2 cosA (cosA +1)}{2cosA (sinA)}\\\\\\:\implies \boxed{ \sf 1 + cos A / sin A.}$

Hence proved .

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