why is this wrong?

[allegansveritatem]

here is the issue and the solution from the book...please bear in mind that this is just a part of the whole problem. This is the denominator of a fraction that is being simplified so that it is composed of factors instead of addends:


Here is what I did:


Why am I wrong here? I realized eventually what the author of the solutions manual did: He first multiplied and then subtracted the result from 1. But why couldn't it have been done the way I did it? I mean , isn't 1-sina/cosb a single term?

On the side: This business of verifying identities really separates the men from the boys. It is, to say truly, nightmarish.

 

[lev888]

here is the issue and the solution from the book...please bear in mind that this is just a part of the whole problem. This is the denominator of a fraction that is being simplified so that it is composed of factors instead of addends:
View attachment 17109

Here is what I did:
View attachment 17110

Why am I wrong here? I realized eventually what the author of the solutions manual did: He first multiplied and then subtracted the result from 1. But why couldn't it have been done the way I did it? I mean , isn't 1-sina/cosb a single term?

On the side: This business of verifying identities really separates the men from the boys. It is, to say truly, nightmarish.

1-sina/cosb is not a thing there. There are no parentheses around it. I don't understand your work. What the book did is simply use cos(a)cos(b) as a common denominator to subtract 2 fractions.

 

[Dr.Peterson]

here is the issue and the solution from the book...please bear in mind that this is just a part of the whole problem. This is the denominator of a fraction that is being simplified so that it is composed of factors instead of addends:
View attachment 17109

Here is what I did:
View attachment 17110

Why am I wrong here? I realized eventually what the author of the solutions manual did: He first multiplied and then subtracted the result from 1. But why couldn't it have been done the way I did it? I mean , isn't 1-sina/cosb a single term?

On the side: This business of verifying identities really separates the men from the boys. It is, to say truly, nightmarish.

To repeat what lev888 said, you are treating it as if it were [MATH]\left(1 - \frac{\sin a}{\cos a}\right)\times \frac{\sin b}{\cos b}[/MATH], when it is really [MATH]1 - \left(\frac{\sin a}{\cos a}\times \frac{\sin b}{\cos b}\right)[/MATH]. Don't forget the order of operations!

Just rewrite it as [MATH]1 - \frac{\sin a\sin b}{\cos a\cos b} = \frac{\cos a\cos b}{\cos a\cos b} - \frac{\sin a\sin b}{\cos a\cos b}[/MATH]

 

[allegansveritatem]

To repeat what lev888 said, you are treating it as if it were [MATH]\left(1 - \frac{\sin a}{\cos a}\right)\times \frac{\sin b}{\cos b}[/MATH], when it is really [MATH]1 - \left(\frac{\sin a}{\cos a}\times \frac{\sin b}{\cos b}\right)[/MATH]. Don't forget the order of operations!

Just rewrite it as [MATH]1 - \frac{\sin a\sin b}{\cos a\cos b} = \frac{\cos a\cos b}{\cos a\cos b} - \frac{\sin a\sin b}{\cos a\cos b}[/MATH]

so, the order of operations....parentheses, exponents, multiply, divide, add, subtract. I guess what threw me here is having spent many days now fooling with trigonometric expressions like 1-sec^2 etc. which are treated as one entity. Then there was the little matter of there being no parenthetical indications. I mean, really, this verifying of identities is enough to drive a fella crazy without taking away his parentheses

 

[Dr.Peterson]

Trig doesn't undo anything you learned in algebra; "1-sec^2" is not treated as one entity. Just keep calm and do the things you've learned all along.

Verifying an identity is tricky mostly just because there are so many things to try. With practice, you can get a sense of what to try first, but largely you just need patience.

 

[allegansveritatem]

1-sina/cosb is not a thing there. There are no parentheses around it. I don't understand your work. What the book did is simply use cos(a)cos(b) as a common denominator to subtract 2 fractions.

well, there were no parentheses around anything so that was an element contributing to my downfall here. That, coupled with a lack of regard on my part for the proper order of operations.....

 

[allegansveritatem]

Trig doesn't undo anything you learned in algebra; "1-sec^2" is not treated as one entity. Just keep calm and do the things you've learned all along.

Verifying an identity is tricky mostly just because there are so many things to try. With practice, you can get a sense of what to try first, but largely you just need patience.

that's it, there are so many things to try....I agree that practice is the only way through here.

 

[Li Batton]

As Dr Peterson states, "Trig doesn't undo anything you learned in algebra." The rules for
adding and subtracting fractions are the same.

Let's say we are subtracting two fractions as follows:

[MATH]\frac{a}{b}-\frac{c}{d}[/MATH]
The common denominator is bd. So we will have

[MATH]\frac{ad-bc}{bd}[/MATH]
In this particular case

[MATH]a=1[/MATH][MATH]b=1[/MATH][MATH]c=sin\alpha sin\beta [/MATH][MATH]d=cos\alpha cos\beta [/MATH]
[MATH]\frac{1}{1}-\frac{sin\alpha sin\beta }{cos\alpha cos\beta }[/MATH]
Which becomes

[MATH]\frac{cos\alpha cos\beta -sin\alpha sin\beta }{cos\alpha cos\beta }[/MATH]