Field extensions contradicting tower law $[\mathbb{R}(x):\mathbb{R}(x+\frac{1}{x})]=2$ since $x$ satisfies a quadratic polynomial over $\mathbb{R}(x+\frac{1}{x})$. Similarly, $[\mathbb{R}(x):\mathbb{R}(x^2+\frac{1}{x^2})]=4$. But, as $(x+\frac{1}{x})^2=(x^2+\frac{1}{x^2})+2$, so 
 $\mathbb{R}(x):\mathbb{R}(x^2+\frac{1}{x^2}):\mathbb{R}(x+\frac{1}{x})$ which clearly contradicts Tower law. What is wrong here? 

I can't see what you are asserting, but here are some facts:
$$|k(x):k(x+1/x)|=2,$$
$$|k(x):k(x^2+1/x^2)|=4,$$
$$|k(x+1/x):k(x^2+1/x^2)|=2.$$
Here $k$ is any field, and $x$ is transcendental over $k$.
