Question

Verify identity

1+ (sin(x/2)/cos(x/2)= (1-cosx-sinx)/sinx

Answer 1

\[1+\frac{ \sin(\frac{ x }{ 2 }) }{ \cos \frac{ x }{ 2 } }= \frac{ 1-cosx+sinx }{ sinx }\]

Answer 2

It's not an identity, how to verify? follow is a counter example:
if x = 90 degree, then x/2 =45, sin (x/2) =cos (x/2) = \(\sqrt2/2\)
the left hand side 1+1 =2
while the right hand side: \(\dfrac{1-\dfrac{\sqrt2}{2}+\dfrac{\sqrt2}{2}}{\dfrac{\sqrt2}{2}}=\frac{1}{\dfrac{\sqrt2}{2}}=\sqrt2\)
You can see the left hand side is 2\(\neq \sqrt2\) is the right hand side
--> It is not an identity.