<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] For $A$, note that
$$
b^2=(\sin a+\cos a)^2=\overbrace{\sin^2a+\cos^2a}^1+2\sin a\cos a
$$
so that
$$
\sin a\cos a=\frac{b^2-1}{2}
$$
A very similar approach works for the rest of expressions.
If you are given the solutions, the best approach is to work backwards: plug in the value of $b$ into the proposed solutions, and simplify them.
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