Question

can we do in any of the one method..is the 1st method correct?!....

Answer 1

You can't solve like method 1. Here, you know that for 45°, sin and cos theta are same. But you can't know the angles always.

Method 2 is the correct way of solving.

in this method, just modify the last step a bit.

$\frac{ \sin( \alpha ) }{ \cos( \alpha ) } = 1 \\ \\ \tan( \alpha ) = 1$

$\tan( \alpha ) = \tan(45) \\ \\ \alpha = 45$

Rest is fine.