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If $\sin x - \cos{x} = \dfrac{1}{3}$, what is $(\sin{x})^3 + (\cos{x})^3$?
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b) Solution:

We need to relate the given information to what is required. Cubing both sides of the equation    won't help because it will ultimately give us ,
which is not what we are required to determine.

We will therefore start by factorising   ,
which is the sum of two cubes.

We can simplify the second set of brackets using the square identity:

Therefore,

In order to find the value of , we will now need to determine the value of   and .

We can relate the two latter expressions through the square identity .

We were given that , so we can use the square identity again, only this time we will square .

We can now determine the value of  :

Finally, we can evaluate   :

Notes: The principles involved in this problem were essentially polynomial multipication, factorisation, and knowledge of trigonometric square identities.  A fair bit of manipulation and some lateral thinking was also required. It goes without saying that the principle of BODMAS needs to be adhered to as well.

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