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So I have an establishing the identity problem that I'm trying to figure out.

$\frac{\sin\theta + \cos\theta} {\sec\theta + \csc\theta} = \cos(\theta) \sin(\theta) $

I'm being told that the first step is to multiply the right side of the identity by

$\frac{\sec\theta + \csc\theta} {\sec\theta + csc\theta} $

because "since this fraction is equivalent to 1, the product is unchanged."

Why are we doing this?

Archaick
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user3370603
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  • Notice the edit – user3370603 Apr 30 '15 at 02:46
  • Because it works quickly. In some ways, it is not a good way to proceed. More natural for a student is to work with the LHS, expressing it as $\frac{\sin\theta+\cos\theta}{\frac{1}{\cos\theta}+\frac{1}{\sin\theta}}$. Then a little algebra takes care of things. Uglier, for sure, but less "magical." – André Nicolas Apr 30 '15 at 02:52
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    Uglier is in the eye of the beholder. There is a simple "reason" for the identity, that $\sin\theta\cos\theta(\sec\theta+\csc\theta)=\sin\theta+\cos\theta$. But as I mention in a comment below, that particular step is "not allowed." – André Nicolas Apr 30 '15 at 03:02
  • I keep thinking this question has been answered before on MSE, but I only find related questions, not the same one. The identity $\frac{\sec\theta+\csc\theta}{\tan\theta + \cot\theta} = \sin\theta + \cos\theta$ (http://math.stackexchange.com/q/171675/139123) is curiously related to this one by the identity $\frac{1}{\tan\theta + \cot\theta} = \sin\theta\cos\theta$. – David K Apr 30 '15 at 03:39

2 Answers2

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If you multiply the right side by $\frac{\sec\theta + \csc\theta}{sec\theta+\csc\theta}$ then you will get $\cos\theta \sin\theta(\frac{\sec\theta + \csc\theta}{\sec\theta+\csc\theta})$. But $\sec\theta = \frac{1}{\cos\theta}$ and $\csc\theta = \frac{1}{\sin\theta}$ so the top of the fraction will simplify to $\sin\theta + \cos\theta$. This is the easiest way to solve the problem in my opinion.

Danny
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  • I wouldn't necessarily think of it as a first step either, but if I were solving the problem I might start with the right hand side as it is simpler to work with. At that point, multiplying by $\frac{\sec\theta + \csc\theta}{\sec\theta + \csc\theta}$ is the logical next step, and from there the problem just works itself out. – Danny Apr 30 '15 at 03:02
  • This approach makes perfect sense. We would like to multiply the RHS by a fraction whose value is $1$ and whose denominator is the same as the denominator of the LHS. – John Joy Apr 30 '15 at 12:39
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I don't know why, but it seems much harder that way. This is a better way:

$$\frac{\sin \theta + \cos\theta}{\sec \theta + \csc\theta} = \frac{\sin \theta + \cos\theta}{\frac{1}{\cos\theta}+\frac{1}{\sin\theta}} = \frac{\sin \theta + \cos\theta}{\frac{\sin \theta + \cos\theta}{\cos\theta \sin\theta}} = \sin\theta \cos\theta$$. This way is much faster and cleaner.

SalmonKiller
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  • Yes, that is how I would have answered it. I don't know where the pulled that from and why that was recommended as the first step. – user3370603 Apr 30 '15 at 02:53
  • @user3370603 are you sure you read the directions right? Cause there might be special instructions or something. – SalmonKiller Apr 30 '15 at 02:53
  • Yes, it told me the first step was "multiplying the right side by $\frac{\sec\theta + \csc\theta} {\sec\theta + csc\theta} $" – user3370603 Apr 30 '15 at 02:55
  • I can tell you why. The easiest way is to multiply both sides of the identity we are looking to prove by $\sec\theta+\csc\theta$. But for no really good reason, there is a high school convention that we must work with one side only. The "trick" is a substitute for the forbidden multiply both sides by $\sec\theta+\csc\theta$. – André Nicolas Apr 30 '15 at 02:56
  • @AndréNicolas it's not a HS convention, it's a convention of proof. However, if you start on one side, you can just divide by $\sec\theta+\csc\theta$ instead of multiplying the other side by it. – SalmonKiller Apr 30 '15 at 02:59
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    In HS trig identities, there is a convention that one does not bother about the issue of reversibility, nor does one bother with where the functions involved are defined. As is often the case, the identity is not really an identity, for the right side is defined everywhere and the left side is not. – André Nicolas Apr 30 '15 at 03:07
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    If $\sec\theta+\csc\theta \neq 0,$ then $\frac{a}{\sec\theta+\csc\theta} = b$ is equivalent to $a = (\sec\theta+\csc\theta)b,$ that is, the multiplication on both sides is reversible. If $\sec\theta+\csc\theta = 0$ then the LHS of the "identity" is undefined, so no proof will work in that case. – David K Apr 30 '15 at 03:33
  • @DavidK I thought that was assumed in the declaration of the identity. We would not need to consider $\sec\theta+\csc\theta = 0$. – SalmonKiller Apr 30 '15 at 03:36
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    @SalmonKiller That is precisely the point. We have to assume $\sec\theta + \csc\theta \neq 0$, so multiplication by that quantity is reversible (that is, division by that quantity is permitted). – David K Apr 30 '15 at 03:48