multiply through with sin and cos

**skoker** · November 6th 2010, 04:09 AM

I am fallowing the solution for a verify identity problem in my textbook, it looks like they multiply through with sin then cos in order to create tan functions under the radical. is this normal method to use? if they are using another method or identity can you explain.

thanks

this is exactly how it is in the book 4 I dont get or step 2.

$ \begin{document} $\sqrt{\frac{2}{1 + \cos \theta} ^{}}$ $\sqrt{\frac{\tmtextit{2 \sin} \theta}{\tmtextit{\sin} \theta \left( 1 + \tmtextit{\cos} \theta \right) }}$ $\sqrt{\frac{\tmtextit{2 \sin} \theta}{\tmtextit{\sin} \theta + \tmtextit{\sin} \theta \tmtextit{\cos} \theta}}$ $\sqrt{\frac{\frac{2 \sin \theta}{\cos \theta}}{ \frac{\tmtextit{\sin} \theta}{\tmtextit{\cos} \theta} + \frac{\tmtextit{\sin} \theta \tmtextit{\cos} \theta}{ \tmtextit{\cos} \theta}}^{}}$ $

**Unknown008** · November 6th 2010, 05:46 AM

I'm not sure what you don't understand, but what you posted is good.

Let's consider only the part within the square root.

$\begin{array}{ccl} \dfrac{2}{1 + \cos\theta} &=& \dfrac{2}{1 + \cos\theta} \times 1 \\ &=& \dfrac{2}{1 + \cos\theta} \times \dfrac{\sin\theta}{\sin\theta} \\ &=& \dfrac{2\sin\theta}{\sin\theta(1 + \cos\theta)} \\ &=& \dfrac{2\sin\theta}{\sin\theta + \sin\theta\cos\theta}\\ &=& \dfrac{2\sin\theta}{\sin\theta + \sin\theta\cos\theta}\times1 \\ &=& \dfrac{2\sin\theta}{\sin\theta + \sin\theta\cos\theta} \times \dfrac{\frac{1}{\cos\theta}}{\frac{1}{\cos\theta}} \\ &=& \dfrac{\frac{2\sin\theta}{\cos\theta}}{\frac{\sin\ theta}{\cos\theta} + \frac{\sin\theta\cos\theta}{\cos\theta}} \end{array}$

**skoker** · November 6th 2010, 06:09 AM

I was not sure if multiplying through with sin or cos in order to advance to the next step was 'legal'. I was looking for a formula or a rule that would introduce sin or cos. but I guess you can just multiply through with trig functions as a general rule, correct?

**Unknown008** · November 6th 2010, 06:14 AM

As I showed you, I only multiplied by 1. This is possible, right?

Well, $\dfrac{\cos\theta}{\cos\theta} = 1$

Hence, there is no rule violated. The same thing applies to sin and other such things.

**skoker** · November 6th 2010, 07:54 AM

I never thought about that formality in this context, 1=x/x. that makes sense thanks.

**Unknown008** · November 6th 2010, 07:57 AM

Yes, those sorts of methods are commonly used in rationalisation, be it concerning trigonometry, or complex numbers, or surds where they are most commonly used.

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