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I have the following identity to verify: $$\sin(x+1)\sin(x+1) - \sin(x+2)\sin x = \sin^2(1).$$

I'm becoming more familiar with sum and difference formulas to some degree, but this one has stumped me.

I don't know if I'm doing it right, even, but I have this so far: $$(\sin x \cos(1) + \cos x \sin(1))^2 - (\sin x \cos(2) + \cos x \sin(2))(\sin x) = \sin^2(1). $$

I don't want to just ask "how i do dis" and expect an answer. I am trying, but my brain doesn't quite understand all this yet.

Please help! I may be late to reply, I have work to get to here.

Thanks a million,

-Jon

Martin Sleziak
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Werewoof
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2 Answers2

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This is an identity you can use: $$\sin(a+b)\sin(a-b)= \sin^2(a)-\sin^2(b)$$

theyaoster
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avz2611
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Using the product to sum formulas, we can rewrite the left side as $$ \frac 12 \left[1 - \cos(2x+2)\right] - \frac 12 \left[\cos(2) - \cos(2x+2)\right] $$ And from there, $\frac 12 [1 - \cos(2 \cdot 1)] = \sin^2(1)$

Ben Grossmann
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  • Thanks for your reply as well! I will study this closely to see what I can glean from it, I really want to understand this and get this in my head as a tool I can reuse. – Werewoof Oct 24 '16 at 19:22