If the equation ${\sin }^4\theta +{\cos }^4\theta =a$ has a real solution then

Solution:

We have,

   $a={\sin }^4\theta +{\cos }^4\theta \le {\sin }^2\theta +{\cos }^2\theta \le 1$

Also, 

     $\begin{array}{l} a={\sin }^4\theta +{\cos }^4\theta ={\left({\sin }^2\theta +{\cos }^2\theta \right)}^2-2{\sin }^2\theta {\cos }^2\theta \\ \Rightarrow a={\sin }^4\theta +{\cos }^{4}0=1-\frac{1}{2}{\sin }^{2}2\theta \\ \Rightarrow {\sin }^{2}20=2(1-a)\Rightarrow 2(1-a)\le 1\Rightarrow a\ge \frac{1}{2}\end{array}$

Hence, $\frac{1}{2}\le a\le 1$