Question

Prove that: √1+cos x/√1-cos x=Cosec x + cot x

Answer 1

see the attachment dude....

jst rationalization....

Answer 2

$\dfrac{ \sqrt{1 \: + \: cos \: x} }{ \sqrt{1 \: - \: cos \: x} }$ = cosec x + cot x

_______________[GIVEN]

• Taking L.H.S.

=> $\dfrac{ \sqrt{1 \: + \: cos \: x} }{ \sqrt{1 \: - \: cos \: x} }$

• Rationalize

=> $\dfrac{ \sqrt{1 \: + \: cos \: x} }{ \sqrt{1 \: - \: cos \: x} }$ × $\dfrac{ \sqrt{1 \: + \: cos \: x} }{ \sqrt{1 \: + \: cos \: x} }$

=> $\dfrac{ { \sqrt{(1 \: + \: cos \: x)} }^{2} }{ \sqrt{sin}^{2}x }$

=> $\dfrac{1\:+\:cos\:x}{sin\:x}$

=> $\dfrac{1}{sin\:x}$ + $\dfrac{sin\:x}{cos\:x}$

=> cosec x + cot x

• L.H.S. = R.H.S.

Hence, proved.

_____________________________

• Used formulas :

cosec x = $\dfrac{1}{sin\:x}$

cot x = $\dfrac{cos\:x}{sin\:x}$

________________________________