Question

The value of $\frac{{\cos }^4\mathrm{\theta }+{\cos }^2\mathrm{\theta }{\sin }^2\mathrm{\theta }+{\sin }^2\mathrm{\theta }}{{\cos }^2\mathrm{\theta }+{\cos }^2\mathrm{\theta }{\sin }^2\mathrm{\theta }+{\sin }^4\mathrm{\theta }}$ is __________.

Answer 1

$$\begin{gathered} \frac{{\cos }^4\theta +{\cos }^2\theta {\sin }^2\theta +{\sin }^2\theta }{{\cos }^2\theta +{\cos }^2\theta {\sin }^2\theta +{\sin }^4\theta } \\ =\frac{{\left({\cos }^2\theta \right)}^2+{\cos }^2\theta {\sin }^2\theta +{\sin }^2\theta }{{\cos }^2\theta +{\cos }^2\theta {\sin }^2\theta +{\sin }^4\theta } \\ =\frac{{\left(1-{\sin }^2\theta \right)}^2+{\cos }^2\theta {\sin }^2\theta +{\sin }^2\theta }{{\cos }^2\theta +{\cos }^2\theta {\sin }^2\theta +{\sin }^4\theta } \\ =\frac{1+{\sin }^4\theta -2{\sin }^2\theta +{\cos }^2\theta {\sin }^2\theta +{\sin }^2\theta }{{\cos }^2\theta +{\cos }^2\theta {\sin }^2\theta +{\sin }^4\theta } \end{gathered}$$
$$\begin{gathered} =\frac{1-{\sin }^2\theta +{\sin }^4\theta +{\cos }^2\theta {\sin }^2\theta }{{\cos }^2\theta +{\cos }^2\theta {\sin }^2\theta +{\sin }^4\theta } \\ =\frac{{\cos }^2\theta +{\cos }^2\theta {\sin }^2\theta +{\sin }^4\theta }{{\cos }^2\theta +{\cos }^2\theta {\sin }^2\theta +{\sin }^4\theta } \\ =1 \end{gathered}$$

The value of $\frac{{\cos }^4\mathrm{\theta }+{\cos }^2{\mathrm{\theta sin}}^2\mathrm{\theta }+{\sin }^2\mathrm{\theta }}{{\cos }^2\mathrm{\theta }+{\cos }^2{\mathrm{\theta sin}}^2\mathrm{\theta }+{\sin }^4\mathrm{\theta }}$ is _____1_____.