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I'm having trouble with the following-

Show: $$\frac{\cos 3x + i\sin 3x}{\cos x + i\sin x} = \cos 2x + i\sin 2x$$

I have tried both multiplying by the complex conjugate of the denominator and alternatively trigonometric identities, but the results don't get close

Can anybody give me any pointers in the right general direction?

Thanks for the help

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  • $\begingroup$ Use hsm.stackexchange.com/questions/4907/… $\endgroup$
    – lab bhattacharjee
    Sep 29, 2017 at 16:20
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    $\begingroup$ Are you supposed to know Euler's formula? $\endgroup$
    – Bernard
    Sep 29, 2017 at 16:22
  • $\begingroup$ Multiplying by the complex conjugate of the denominator will work. If you "didn't get close" then your idea is correct, but you have an error in execution. Check your algebra. $\endgroup$
    – Doug M
    Sep 29, 2017 at 16:43

2 Answers 2

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Hint: $$e^{i ax} = \cos(ax) + i \sin(ax).$$

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  • $\begingroup$ Thank you, I have understand it now :) $\endgroup$
    – user485898
    Sep 29, 2017 at 16:24
  • $\begingroup$ Thanks everyone for answers $\endgroup$
    – user485898
    Sep 29, 2017 at 16:24
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HINT: multiply numerator and denominator by $$\cos(x)-i\sin(x)$$ you will get $$\cos(x)\cos(3x)+\sin(x)\sin(3x)+i(\cos(x)\sin(3x)-\cos(3x)\sin(x))$$ and this is for what you are searching

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