Trigonometry#1

$\cos^2 48^\circ - \sin^2 12^\circ + \sin 132^\circ \sin 12^\circ$

$= \cos^2 48^\circ - \sin^2 12^\circ + \cos(90^\circ - 132^\circ) \sin 12^\circ$

$= \cos^2 48^\circ - \sin^2 12^\circ + \cos(-42^\circ) \sin 12^\circ$

$= \cos^2 48^\circ - \sin^2 12^\circ + \cos(42^\circ) \sin 12^\circ$

$= \cos^2 48^\circ - \sin^2 12^\circ + \dfrac{\sin (42^\circ + 12^\circ) - \sin (42^\circ - 12^\circ)}{2}$

$= \cos^2 48^\circ - \sin^2 12^\circ + \dfrac{\sin (54^\circ) - \sin (30^\circ)}{2}$

$= \cos^2 48^\circ - \sin^2 12^\circ + \dfrac{\sin (54^\circ) - \frac{1}{2}}{2}$

$= \cos^2 48^\circ - \sin^2 12^\circ + \dfrac{2\sin (54^\circ) - 1}{4}$

$= \cos^2 48^\circ - \sin^2 12^\circ + \dfrac{2\cos (90^\circ- 54^\circ) - 1}{4}$

$= \cos^2 48^\circ - \sin^2 12^\circ + \dfrac{2\cos (36^\circ) - 1}{4}$

$= \cos^2 48^\circ - \sin^2 12^\circ + \dfrac{2 \times \frac{\sqrt{5} + 1}{4} - 1}{4}$ $~~~~~~~~~~~~~~~~~~$ $; \left[\cos 36^\circ = \dfrac{\sqrt{5}+1}{4} \right]$

After simplifying, we get

$\Rightarrow \cos^2 48^\circ - \sin^2 12^\circ + \dfrac{\sqrt{5} - 1}{8} \approx 0.5590$