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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: $A\:=\:\frac{1}{2}\left(b^2-1\right),\:B\:=\sqrt{2-b^2},C=\left|b\right|\sqrt{2-b^2},\:D\:=\:\frac{\left|b\right|}{2}\left(3-b^2\right),\:E=\frac{1}{2}\left(1+2b^2-b^4\right)$

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    You have your own answer. Just substitute them. – 2017-01-04
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    It is given, I do not know how to get it. – 2017-01-04
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    Compute square of $b$. – 2017-01-04

2 Answers 2

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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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Hints

$A)$

$$(\sin a +\cos a)^2=1+2\sin a \cos a$$

$B)$

$$(\sin a - \cos a)^2=1-2\sin a \cos a$$

$C)$

$$\sin^2a-\cos^2a=(\sin a+\cos a)(\sin a - \cos a)$$

$D)$

$$\sin^3a+\cos ^3a=(\sin a +\cos a)(1-\sin a \cos a)$$

$E)$

$$\sin^4 a+\cos^4a=(\sin^2a+\cos^2 a)^2-2(\sin a\cos a)^2=1-2(\sin a\cos a)^2$$