$\cos{x} - \sec{x} = \frac{3}{2}$ হলে cos⁴x+sin⁴x এর মান কত?

$\begin{aligned} & \cos x-\frac{1}{\cos x}=\frac{3}{2} \\ \Rightarrow & \frac{\cos ^{2} x-1}{\cos x}=\frac{3}{2}\end{aligned}$

$\begin{array}{l}\Rightarrow 2 \cos ^{2} x-2=3 \cos x \\ \Rightarrow 2 \cos ^{2} x-4 \cos x+\cos x-2=0 \\ \Rightarrow 2 \cos x(\cos x-2)+1(\cos x-2)=0 \\ \cos x \neq 2, \quad 2 \cos x=-1 \\ \cos x=-\frac{1}{2} \\ x=120^{\circ}\end{array}$

$\begin{aligned} \cos ^{4}\left(120^{\circ}\right)+\sin ^{4}\left(120^{\circ}\right) & =\frac{1}{16}+\frac{9}{16} \\ & =\frac{10}{16}=\frac{5}{8}\end{aligned}$