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Prove that $$ \lim_{\beta\to\alpha}\;\frac{\alpha\sin\beta-\beta\sin\alpha}{\alpha \cos \beta- \beta \cos \alpha}= \tan(\alpha-\tan^{-1}\alpha) $$

I am solving the exercise from the S.L. Loney plane trigonometry book, page 48 question number 35, I got stuck.

Any help will be truly appreciated.

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  • $\begingroup$ L’Hôpital’s rule? $\endgroup$
    – Tavish
    May 4 at 19:21
  • $\begingroup$ I used series expansion and tried to do it. We are not allowed to use derivatives. $\endgroup$
    – Sayantan
    May 4 at 19:27
  • $\begingroup$ Do you mean the limit as $\beta$ tends to $a$? $\endgroup$ May 4 at 19:35
  • $\begingroup$ When you got the limit, you can derive the required formula using $$tan(x-y)= \frac{tanx-tany}{1+tanx \cdot tany}$$ $\endgroup$
    – ThomasL
    May 4 at 20:01
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Hint:

Write the numerator as

$$a(\sin \beta-\sin a)-(\beta-a)\sin a=(\beta-a)\left(a\cdot\dfrac{\sin\beta-\sin a}{\beta -a}-\sin a\right)$$

Similarly for the denominator

Then use https://mathworld.wolfram.com/ProsthaphaeresisFormulas.html

Alternatively, replace $\beta$ with $a+h$ to find the numerator $$=a\sin(a+h)-(a+h)\sin a=a(\sin(a+h)-\sin a)-h\sin a$$

Similarly for the denominator

Now divide the numerator and the denominator by $h$ as $h\ne0$ as $h\to0$

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Considering $$\frac{\alpha\sin(\beta)-\beta\sin(\alpha)}{\alpha \cos (\beta)- \beta \cos (\alpha)}$$ make life simpler letting $\beta=\alpha+x$ to face the problem of the limit, when $x\to 0$ of $$\frac{\sin (\alpha ) (\alpha +x)-\alpha \sin (\alpha +x)}{\cos (\alpha ) (\alpha +x)-\alpha \cos (\alpha +x)}$$ Expanding, numerator and denominator $$\text{num}=\alpha \sin (\alpha )+x \sin (\alpha )-\alpha \cos (\alpha ) \sin (x)-\alpha \sin (\alpha ) \cos (x)$$ $$\text{den}=\alpha \cos (\alpha )+\alpha \sin (\alpha ) \sin (x)+x \cos (\alpha )-\alpha \cos (\alpha ) \cos (x)$$

Now, use Taylor series or, simpler, equivalents to make $$\frac{\text{num}}{\text{den}}\sim \frac {x (\sin (\alpha )-\alpha \cos (\alpha )) } {x (\alpha \sin (\alpha )+\cos (\alpha )) }=\frac{\sin (\alpha )-\alpha \cos (\alpha )}{\alpha \sin (\alpha )+\cos (\alpha )}=\frac{ \tan (\alpha )-\alpha}{1+\alpha \tan (\alpha ) }$$ and recognize a well known trigonometric formula.

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