a) \(\frac{1}{\sqrt{3}+\sqrt{2}}\) + \(\sqrt{7-4\sqrt{3}}\) +\(\sqrt{2}\)
= \(\frac{\sqrt{3}-\sqrt{2}}{3-2}\) + \(\sqrt{4-2.2.\sqrt{3}+3}\)+\(\sqrt{2}\)
=\(\sqrt{3}-\sqrt{2}\) + \(\sqrt{\left(2-\sqrt{3}\right)^2}\)+\(\sqrt{2}\)
= \(\sqrt{3}-\sqrt{2}\)+ \(\left|2-\sqrt{3}\right|\)+\(\sqrt{2}\)
= \(\sqrt{3}-\sqrt{2}\) + 2-\(\sqrt{3}\) + \(\sqrt{2}\)
=2.
b) \(\left(1+\frac{1}{\cot^220^o}\right)\). \(\cot^220^o\)- \(\tan40^o.\tan50^o\)
=\(\cot^220^o\) + 1 - \(\tan40^o\) . \(\cot40^o\)
=\(\cot^220^o\) + 1-1
= \(\cot^220^o\).
c) A= \(\frac{\sqrt{x}-1}{2\sqrt{x}+1}\) - \(\frac{3}{1-2\sqrt{x}}\) - \(\frac{4\sqrt{x}+4}{4x-1}\) , ĐK: x ≥ \(\frac{1}{4}\)
= \(\frac{\sqrt{x}-1}{2\sqrt{x}+1}\) +\(\frac{3}{2\sqrt{x}-1}\) - \(\frac{4\sqrt{x}+4}{\left(2\sqrt{x}+1\right)\left(2\sqrt{x}-1\right)}\)
= \(\frac{\left(\sqrt{x}-1\right)\left(2\sqrt{x}-1\right)}{\left(2\sqrt{x}+1\right)\left(2\sqrt{x}-1\right)}\) + \(\frac{3\left(2\sqrt{x}+1\right)}{\left(2\sqrt{x}+1\right)\left(2\sqrt{x}-1\right)}\)- \(\frac{4\sqrt{x}+4}{\left(2\sqrt{x}+1\right)\left(2\sqrt{x}-1\right)}\)
=\(\frac{\left(\sqrt{x}-1\right)\left(2\sqrt{x}-1\right)+3\left(2\sqrt{x}+1\right)-4\sqrt{x}-4}{\left(2\sqrt{x}+1\right)\left(2\sqrt{x}-1\right)}\)
= \(\frac{2x-\sqrt{x}-2\sqrt{x}+1+6\sqrt{x}+3-4\sqrt{x}-4}{\left(2\sqrt{x}+1\right)\left(2\sqrt{x}-1\right)}\)
= \(\frac{2x-\sqrt{x}}{\left(2\sqrt{x}+1\right)\left(2\sqrt{x}-1\right)}\)
= \(\frac{\sqrt{x}\left(2\sqrt{x}-1\right)}{\left(2\sqrt{x}+1\right)\left(2\sqrt{x}-1\right)}\)
= \(\frac{\sqrt{x}}{2\sqrt{x}+1}\).
