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given the identity sin(x+y)=sinx cosy + siny cosx
sin2x = 2 sinx cosx and
sin(2(x)+x) = sin 2x cos x + sinx cos 2x
using the last two identities gives
sin3x= 2 sinx cosx cosx + sinx cos2x
factoring the sinx we have
sin3x = sinx(2cosx cosx+cos2x)
which satisfies the requirement.
However, we can simplify further since cos 2x = cosx cosx - sinx sinx (a well known identity)
sin3x = sinx (2cosx cosx +cosx cosx - sinx sinx)
so sin3x= sinx(3cosx cosx - sinx sinx)
or sin 3x = 3.cos²x.sinx - sin³x
* * * * *
Good, but not good enough. The answer was required in terms of sin, not a mixture of sinx and cosx. Easily recitified, though, since cos²x = 1 - sin²x
Therefore sin3x = 3*(1-sin²x)*sinx - sin³x
= 3sinx - 4sin³x
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Are 3x and x like terms?
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What is x minus 4x?
The expression "x minus 4x" can be simplified by combining like terms. When subtracting 4x from x, we are essentially subtracting 4 of the x terms from 1 of the x terms, resulting in a final answer of -3x. So, x minus 4x simplifies to -3x.
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