1
$\begingroup$

How would I solve the following double angle identity. $$ \frac{\sin(A+B)}{\cos(A-B)}=\frac{\tan A+\tan B}{1+\tan A\tan B} $$ So far my work has been. $$ \frac{\sin A\cos B+\cos A\sin B}{\cos A\cos B+\sin A\sin B} $$ But what would I do to continue.

  • 2
    Divide numerator and denominator by $\cos A \cos B$.2012-07-25
  • 0
    Oh I see now dividing by cos I get the correct answer thanks to all who posted.2012-07-25
  • 3
    One can _prove_ and _identity_ or _solve_ an _equation_. But to speak of _solving_ an _identity_ could leave some doubt about what you mean.2012-07-25
  • 0
    @Rick Decker: please do not change the variables from $A,B$ to $x,y$ as avatar's comment and my answer were in terms of $A,B$2012-07-25
  • 0
    @Ross. Sorry about that. avatar's comment, your answer, and my edit came virtually on top of each other; I didn't see the notifications while I was editing. Note to self: for questions that are likely to be answered immediately after they're posted, delay editing until the dust settles.2012-07-25
  • 0
    @RickDecker: No big problem. I put it back.2012-07-25

2 Answers 2

5

Now divide by $\cos A \cos B$ and you are there

1

What i get is, how to solve the problem?? Is that correct then here u are:

$$\dfrac{\dfrac{\sin x\cos y + \cos x\sin y}{\cos x\cos y}} {\dfrac{\cos x\cos y + \sin x\sin y}{\cos x\cos y}}$$

$$\dfrac{\dfrac{\sin x\cos y}{\cos x\cos y} +\dfrac{\cos x\sin y}{\cos x\cos y}} {\dfrac{\cos x\cos y} {\cos x\cos y}+ \dfrac{\sin x\sin y}{\cos x\cos y}}$$ $$\dfrac{\tan x+\tan y}{1+\tan x\tan y}$$