If \(yz=a^2\) prove that \[ \frac{1}{a+y} + \frac{1}{a+z} = \frac{1}{a}.\]

By using this result, or otherwise, find the numerical values of

- \(\dfrac{1}{1+x^k} + \dfrac{1}{1+x^{-k}}\);
- \(\dfrac{1}{7+\sqrt{62}-\sqrt{13}} + \dfrac{1}{7+\sqrt{62}+\sqrt{13}}\);
- \(\dfrac{2}{5(2+5\log_b c)} + \dfrac{1}{5+2\log_c b}\);