(1) 化简：$\left(\frac{a}{a+b}-\frac{a^2}{a^2+2ab+b^2}\right)\div\left(\frac{a}{a+b}-\frac{a^2}{a^2-b^2}\right)$. (2) 计算：$\frac{1}{2}\lg 25+\lg 2-\lg \sqrt{0.1}-\log_2 9\times\log_3 2$. (3) $\sqrt{-1}=\mathrm{i}$，验算 $\mathrm{i}$ 是否方程 $2x^4+3x^3-3x^2+3x-5=0$ 的解. (4) 求证：$\frac{\sin\left(\frac{\pi}{4}+\theta\right)}{\sin\left(\frac{\pi}{4}-\theta\right)}+\frac{\cos\left(\frac{\pi}{4}+\theta\right)}{\cos\left(\frac{\pi}{4}-\theta\right)}=\frac{2}{\cos 2\theta}$.