Solve the equationsec x - sec x sin^{2}x = cos x

I calculated ${\displaystyle {\sin 75}^{\circ }}$ as ${\displaystyle \frac{1}{2\sqrt{2}}+\frac{\sqrt{3}}{2\sqrt{2}}}$, but the answer is ${\displaystyle \frac{\sqrt{2}+\sqrt{6}}{4}}$. What went wrong?
I calculated the exact value of ${\displaystyle {\sin 75}^{\circ }}$ as follows:
${\displaystyle {\sin 75}^{\circ }=\sin ({30}^{\circ }+{45}^{\circ })}$
$=\sin 30°\cos 45°+\cos 30°\sin 45°$
$=\frac{1}{2}·\frac{1}{\sqrt{2}}+\frac{\sqrt{3}}{2}·\frac{1}{\sqrt{2}}$
$=\frac{1}{2\sqrt{2}}+\frac{\sqrt{3}}{2\sqrt{2}}$
The actual answer is
${\displaystyle \frac{\sqrt{2}+\sqrt{6}}{4}}$