Question

Prove that $se{c}^{2}x+co{s}^{2}x\ge 2$

Answer 1

$\begin{array}{cc}co{s}^{2}x\ne 0& \left(1\right)se{c}^{2}x+co{s}^{2}x\ge 2\\ cos\frac{1}{0}isnotdefined.& \Rightarrow \frac{1}{co{s}^{2}x}+co{s}^{2}x\\ \Rightarrow x\ne (2n+1)\frac{\pi }{2}& AM\ge GM\\ & \Rightarrow \frac{\frac{1}{co{s}^{2}x}+co{s}^{2}x}{2}\ge \sqrt{1}\\ & \Rightarrow \frac{1}{co{s}^{2}x}+co{s}^{2}x\ge 2\\ & forallxexcept(2x+1)\frac{\pi }{2}\end{array}$