a) Korzystamy ze wzoru:

$\sin 2\alpha =2\sin \alpha \cos \alpha$  

$\sin \left(2\cdot {22}^{\circ }{30}^{{}^{\prime }}\right)=2\sin {22}^{\circ }{30}^{{}^{\prime }}\cos {22}^{\circ }{30}^{{}^{\prime }}$      

$\sin {45}^{\circ }=2\sin {22}^{\circ }{30}^{{}^{\prime }}\cos {22}^{\circ }{30}^{{}^{\prime }}$  

$\frac{\sqrt{2}}{2}=2\sin {22}^{\circ }{30}^{{}^{\prime }}\cos {22}^{\circ }{30}^{{}^{\prime }}\mid :2$  

$\frac{\sqrt{2}}{4}=\sin {22}^{\circ }{30}^{{}^{\prime }}\cos {22}^{\circ }{30}^{{}^{\prime }}$  

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b) Zauważmy, że:

$-\cos \pi \cdot {\cos }^2\frac{\pi }{12}-2{\sin }^2\frac{\pi }{12}\cdot \sin \frac{\pi }{6}=$  

$=-\left(-1\right)\cdot {\cos }^2\frac{\pi }{12}-2{\sin }^2\frac{\pi }{12}\cdot \frac{1}{2}={\cos }^2\frac{\pi }{12}-{\sin }^2\frac{\pi }{12}$  

Korzystamy ze wzoru:

$\cos 2\alpha ={\cos }^2\alpha -{\sin }^2\alpha$  

$\cos \left(2\cdot \frac{\pi }{12}\right)={\cos }^2\frac{\pi }{12}-{\sin }^2\frac{\pi }{12}$  

$\cos \frac{\pi }{6}={\cos }^2\frac{\pi }{12}-{\sin }^2\frac{\pi }{12}$  

$\frac{\sqrt{3}}{2}={\cos }^2\frac{\pi }{12}-{\sin }^2\frac{\pi }{12}$  

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c) Zauważmy, że:

$4\sin {15}^{\circ }{\cos }^3{15}^{\circ }-4{\sin }^3{15}^{\circ }\cdot \cos {15}^{\circ }=4\sin {15}^{\circ }\cos {15}^{\circ }\left({\cos }^2{15}^{\circ }-{\sin }^{\circ }{15}^{\circ }\right)=$  

$=2\cdot 2\sin {15}^{\circ }\cos {15}^{\circ }\cdot \cos \left(2\cdot {15}^{\circ }\right)=2\cdot \sin \left(2\cdot {15}^{\circ }\right)\cdot \cos {30}^{\circ }=$  

$=2\cdot \sin {30}^{\circ }\cdot \cos {30}^{\circ }=2\cdot \frac{1}{2}\cdot \frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{2}$  

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d) Zauważmy, że:

${\cos }^4{7}^{\circ }{30}^{{}^{\prime }}-6{\sin }^2{7}^{\circ }{30}^{{}^{\prime }}\cdot {\cos }^2{7}^{\circ }{30}^{{}^{\prime }}+{\sin }^4{7}^{\circ }{30}^{{}^{\prime }}=$  

$={\left({\sin }^2{7}^{\circ }{30}^{{}^{\prime }}+{\cos }^2{7}^{\circ }{30}^{{}^{\prime }}\right)}^2-8{\sin }^2{7}^{\circ }{30}^{{}^{\prime }}{\cos }^2{7}^{\circ }{30}^{{}^{\prime }}=$  

$=1^2-2\cdot 4{\sin }^2{7}^{\circ }{30}^{{}^{\prime }}{\cos }^2{7}^{\circ }{30}^{{}^{\prime }}=1-2{\left(2\sin 7^{\circ }{30}^{{}^{\prime }}\cos 7^{\circ }{30}^{{}^{\prime }}\right)}^2=$  

$=1-2{\left(\sin \left(2\cdot 7^{\circ }{30}^{{}^{\prime }}\right)\right)}^2=1-2{\sin }^2{15}^{\circ }=\cos \left(2\cdot {15}^{\circ }\right)=\cos {30}^{\circ }=\frac{\sqrt{3}}{2}$