<s>[INST] Trigonometric functions I need some help with these tasks and would be very grateful if someone shows me the way of solving them. 
If $\sin\left(a\right)+\cos\left(a\right)=b$ and $\left|b\right|\le \sqrt{2}$, represent the following expressions with $b$:
$A=\sin\left(a\right)\cdot \cos\left(a\right)$,
$B=\left|\sin\left(a\right)-\cos\left(a\right)\right|$,
$C=\left|\sin^2\left(a\right)-\cos^2\left(a\right)\right|$,
$D=\left|\sin^3\left(a\right)+\cos^3\left(a\right)\right|$,
$E=\sin^4\left(a\right)+\cos^4\left(a\right)$
Answer: $A\:=\:\frac{1}{2}\left(b^2-1\right),\:B\:=\sqrt{2-b^2},C=\left|b\right|\sqrt{2-b^2},\:D\:=\:\frac{\left|b\right|}{2}\left(3-b^2\right),\:E=\frac{1}{2}\left(1+2b^2-b^4\right)$
 [/INST] Hints
$A)$
$$(\sin a +\cos a)^2=1+2\sin a \cos a$$
$B)$ 
$$(\sin a - \cos a)^2=1-2\sin a \cos a$$
$C)$
$$\sin^2a-\cos^2a=(\sin a+\cos a)(\sin a - \cos a)$$
$D)$
$$\sin^3a+\cos ^3a=(\sin a +\cos a)(1-\sin a \cos a)$$
$E)$
$$\sin^4 a+\cos^4a=(\sin^2a+\cos^2 a)^2-2(\sin a\cos a)^2=1-2(\sin a\cos a)^2$$
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