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$\frac{\sin \left(2k+2\right)\theta -\sin \left(2k\right)\theta }{2sin\theta }=cos\left(2k+1\right)\theta $

How do I show that the above is equal to each other?

The problem is part of an induction question and i am unable to show that LHS=RHS

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    Use http://mathworld.wolfram.com/ProsthaphaeresisFormulas.html – 2017-02-07
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    Use sum to product formula $$\sin a - \sin b =2 \sin (\dfrac{a-b}{2}) \cos (\dfrac{a+b}{2})$$ – 2017-02-07
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    Thank you, I understand now. – 2017-02-07

2 Answers 2

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We know that $$\sin \alpha - \sin \beta = 2 \cos \frac {\alpha + \beta}{2} \sin \frac {\alpha - \beta}{2} $$

Now seeking $\alpha = (2k+2)\theta $ and $\beta = 2k\theta $ and simplifying, the result follows. Hope it helps.

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Using -

$\sin A - \sin B = 2 \cos \frac{A+B}{2} \sin \frac{A-B}{2}$

So from above question we have -

$$= \frac{2 \cos \frac{(2k+2+2k)\theta}{2} \sin \frac{(2k+2-2k)\theta}{2}}{2\sin \theta}$$

$$= \frac{2 \cos (2k+1)\theta \sin \theta}{2\sin \theta}$$

$$=\cos (2k+1)\theta$$

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    Yes typo. Thnk u. – 2017-02-07
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    @Rohan I got to 2cos(2k+1)θsinθ2sinθ step but I am not sure how to simplify it to become cos(2k+1)θ – 2017-02-07
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    Eliminate $2\sin\theta$ in numerator and denominator. – 2017-02-07
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    You left with $\cos(2k+1)\theta$. Equals to right hand side. – 2017-02-07
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    @SonJerm Hope you know that $\sin$ and $\theta $ are connected. So don't cancel them separately. Now coming to the point, you can see that the $2$ factor cancels out and then $\sin \theta $ factor also cancels out. Now what you get is $\cos (2k+1)\theta $. – 2017-02-07
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    Thank you very much @Rohan and Kanawaljit Singh I now understand how to show two identities are identical using the sum to product formula. – 2017-02-07
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    @Rohan Sorry but one more question relating to this topic, what would sin(2k) theta be identical to? I know that sin2 theta is 2sin theta cos theta – 2017-02-07
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    @SonJerm It is equal to $2\sin k\theta \cos k\theta $. Just check it for few values of $k $ if you want to get satisfied. – 2017-02-07
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    In this formula angle becomes half. So $\sin 2k\theta$ becomes $2\sin k\theta \cos k\theta$ – 2017-02-07