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For 4.3.40 to 4.3.42 of my copy of abramowitz and stegun, the relationships between squares of sines and cosines was discussed. It provide the following formulas:

$$\sin^2(x)-\sin^2(y)=\sin(x + y)\sin(x - y)$$ $$\cos^2(x)-\cos^2(y)=-\sin(x + y)\sin(x - y)$$ $$\cos^2(x)-\sin^2(y)=\cos(x + y)\cos(x - y)$$

What is for $\sin^2(x)-\cos^2(y)$? And how did these formulas have been derived?

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up vote 5 down vote accepted

Hint: Use the identity

$$ \sin^2 x+\cos^2 x =1 .$$

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@Downvoter: Why just this rush to downvote answers? – Mhenni Benghorbal Dec 20 '13 at 9:20
You think in a very advance manner. This was the last thing I used to solve my problem. – Ralf17 Dec 20 '13 at 9:31
@Ralf17: Good job. – Mhenni Benghorbal Dec 20 '13 at 9:59
Thanks for your consideration Dr. :+) – Babak S. Dec 20 '13 at 10:34
@B.S.: Thanks for the comment. – Mhenni Benghorbal Dec 20 '13 at 10:36

$\sin^2(x) - \cos^2(y) = - (\cos^2(y)-\sin^2(x)) = \ ?$

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I was thinking about that possibility before I asked the question, but I am looking for a longer answer. Like the manner in which the quadratic formula was derived. – Ralf17 Dec 20 '13 at 9:15
I'm not sure what you're asking. Is it not clear what the corresponding formula should be from what I wrote? – ronno Dec 20 '13 at 9:26
Actually all of your answers were right. I was asking two answers though. It is true that you can get the left part for solving the right part through the addition and subtraction formulas for sine and cosine and by the pythagorean theorem. – Ralf17 Dec 20 '13 at 9:30

The first question is answered by ronno. And, for your second question, you can see that these are true just by simplifying the right hand side of each equation.

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If one accepts these three identities: $$ \sin^2\theta + \cos^2\theta=1 $$ $$ \sin(x+y)=\sin x \cos y + \cos x \sin y $$ $$ \cos(x+y)=\cos x \cos y - \sin x \sin y $$ Then a large class of other identities follows, including the ones in your question.

Now why would a person accept the above three identities? I don't know of their historical proofs although the first is usually attributed to pythagoras. The way one would go about proving these identities depends on the way one $\textbf{defines}$ the concepts of $\sin$ and $\cos$.

My preferred definition involves infinite series and the methods I would use to prove these identities rely on ideas from analysis (which includes calculus) and the use of complex numbers. This is most definitely not the way that these identities were historically proved but I feel it to be a more fundamental way of understanding them.

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These can be established if $\sin$ and $\cos$ are defined in terms of $e$. – Shaun Dec 20 '13 at 10:50

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