Question

prove LHS= RHS, please tell me where I'm wrong​

Answer 1

Answer:

The mistake is rectified in the attached image

Answer 2

Answer:

Your first step for rationalization is partially incorrect.

It is as follows:

$\frac{1}{1+\sqrt{2} }*\frac{1-\sqrt{2} }{1-\sqrt{2} } \\\\=\frac{1-\sqrt{2} }{1-(\sqrt{2})^{2} } \\\\=\frac{1-\sqrt{2} }{1-2} } \\\\=\frac{1-\sqrt{2} }{-1} } \\\\= \sqrt{2} }{-1}$

Therefore your final step will be:

   $\sqrt{2} -1 +\sqrt{3} -\sqrt{2}$

$= \sqrt{3}-1$