$\text{a) }\sin \alpha \cdot \cos \alpha$  

Mamy

$\sin \alpha +\cos \alpha =\alpha {\mid }^{\left(2\right)}$

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

$1+2\sin \alpha \cos \alpha ={\alpha }^2\mid -1$   

$2\sin \alpha \cos \alpha ={\alpha }^2-1\mid :2$  

$\sin \alpha \cos \alpha =\frac{{\alpha }^2-1}{2}$  

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$\text{b) }\sin {\alpha }^3+{\cos }^3\alpha$  

Mamy

$\sin \alpha +\cos \alpha =\alpha {\mid }^{\left(2\right)}$

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

$1+2\sin \alpha \cos \alpha ={\alpha }^2\mid -1$   

$2\sin \alpha \cos \alpha ={\alpha }^2-1\mid :2$  

$\sin \alpha \cos \alpha =\frac{{\alpha }^2-1}{2}$  

Oraz

$\sin {\alpha }^3+{\cos }^3\alpha =\left(\sin \alpha +\cos \alpha \right)\left({\sin }^2\alpha -\sin \alpha \cos \alpha +{\cos }^2\alpha \right)=\alpha \left(1-\frac{{\alpha }^2-1}{2}\right)=\frac{\alpha \left(-{\alpha }^2+3\right)}{2}$  

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$\text{c) }{\sin }^4\alpha +{\cos }^4\alpha$  

Mamy

$\sin \alpha +\cos \alpha =\alpha {\mid }^{\left(2\right)}$

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

$1+2\sin \alpha \cos \alpha ={\alpha }^2\mid -1$   

$2\sin \alpha \cos \alpha ={\alpha }^2-1\mid :2$  

$\sin \alpha \cos \alpha =\frac{{\alpha }^2-1}{2}$  

Zatem

$\sin \alpha +\cos \alpha =\alpha {\mid }^{\left(4\right)}$

${\sin }^4\alpha +4{\sin }^3\alpha \cos \alpha +6{\sin }^2\alpha {\cos }^2\alpha +4\sin \alpha {\cos }^3\alpha +{\cos }^4\alpha ={\alpha }^4\mid -6{\sin }^2\alpha {\cos }^2\alpha$   

${\sin }^4\alpha +{\cos }^4\alpha +4\sin \alpha \cos \alpha \left(\sin \alpha +\cos \alpha \right)={\alpha }^4-6{\sin }^2\alpha {\cos }^2\alpha \mid -4\sin \alpha \cos \alpha \left({\sin }^2\alpha +{\cos }^2\alpha \right)$   

${\sin }^4\alpha +{\cos }^4\alpha ={\alpha }^4-6\cdot {\left(\sin \alpha \cos \alpha \right)}^2-4\sin \alpha \cos \alpha$   

${\sin }^4\alpha +{\cos }^4\alpha ={\alpha }^4-6\cdot {\left(\frac{{\alpha }^2-1}{2}\right)}^2-4\cdot \frac{{\alpha }^2-1}{2}$   

  ${\sin }^4\alpha +{\cos }^4\alpha ={\alpha }^4-6\cdot \frac{{\alpha }^4-2{\alpha }^2+1}{4}-2{\alpha }^2+2$   

  ${\sin }^4\alpha +{\cos }^4\alpha ={\alpha }^4+\frac{-3{\alpha }^4+6{\alpha }^2-3}{2}-2{\alpha }^2+2$   

  ${\sin }^4\alpha +{\cos }^4\alpha =\frac{2{\alpha }^4-3{\alpha }^4+6{\alpha }^2-3-4{\alpha }^2+4}{2}$   

  ${\sin }^4\alpha +{\cos }^4\alpha =\frac{-{\alpha }^4+2{\alpha }^2+1}{2}$