Question

solve the question present in the image.
WRONG ANSWER WILL BE REPORTED INSTANTLY!!!!!!!​

Answer 1

LHS

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

Rationalising the denominator

= $\sqrt{ \frac{1 - cos \: A }{1 + cos \: A} \times \frac{1 - cos \: A}{1 - cos \: A} }$

= $\sqrt{ \frac{ {(1 - cos \: A )\: }^{2} }{ {1}^{2} - {cos}^{2} A} }$

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

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

= $cosec \: A - cot \: A$

= RHS

Hence, proved

Answer 2

$\bigstar\huge\bf\underline{\underline{Answer:-}}$

$\bigstar\bf\underline{\underline{To \ prove:-}}$

$\bf \sqrt{\frac{1-cosA}{1+cosA}}=cosecA - cotA$

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

$\bf LHS = \sqrt{\frac{1-cosA}{1+cosA}}=cosecA - cotA$

$\implies\bf\sqrt{\frac{1-cosA}{1+cosA}\times\frac{1-cosA}{1-cosA}}$

$\implies\bf\sqrt{\frac{(1+cosA)^{2}}{1-cos^{2}A}}$

$\implies\bf\sqrt{\frac{(1+cosA)^{2}}{sin^{2}A}}$

$\implies\bf\frac{1-cosA}{sin^{2}A}$

$\implies\bf\frac{1-cosA}{sinA}$

$\implies\boxed{\boxed{\bf cosecA - cotA}}$