Question

Solve:-


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Answer 1

$\huge\underline{\overline{\mid{\bold{\blue{\mathcal{ANSWER}}\mid}}}}$

Given:-

$\frac{cos A - sin A + 1}{cos A + sin A - 1} = cosec A + cot A$

$L.H.S = \frac{cos A - sin A + 1}{cos A + sin A - 1}$

$R.H.S = cosec A + cot A$

To Prove:-

$\frac{cos A - sin A + 1}{cos A + sin A - 1} = cosec A + cot A$

Proof:-

Let us Solve the L.H.S to solve this

$L.H.S = \frac{cos A - sin A + 1}{cos A + sin A - 1}$

Divide Numerator and Denominator by Sin A

$= > \frac{ \frac{cos A - sin A + 1}{sin \: A} }{ \frac{cos A + sin A - 1}{sin A } }$

$= > \frac{ \frac{cos A }{sin A } - \frac{sin A }{sin A } + \frac{1}{sin A } }{ \frac{cos A }{sin A } + \frac{sin A }{sin A } - \frac{1}{sin A } }$

Now, we will use Trigonometric ratios

$= > \frac{cot A - 1 +cosecA }{cot A + 1 - cosecA}$

We can write :

$1 = {cosec}^{2} A + {cot}^{2} A$

as this is a Trigonometry identity

So,

$= > \frac{cot A + cosecA - ( {cosec}^{2}A - {cot}^{2}A)}{cot A + 1 - cosecA}$

$= > \frac{cot A + cosecA - ( cosecA - cotA)(cosecA - cotA)}{cot A + 1 - cosecA}$

$= > \frac{(cot A + cosecA )( 1 - cosecA + cotA)}{cot A + 1 - cosecA}$

$= > cot A + cosecA$

$L.H.S = cot A + cosecA$

Now,

L.H.S=R.H.S

HENCE , PROVED