Proving an existence using mean value theorem.

Question

Exact statement of problem from the book:

If $0<a<b<(π/2)$, using Cauchy's mean value theorem, prove that $(\sin(a) - \sin(b))/(\cos(b) - \cos(a)) = \cot(x)$ for some $x$ in $(a,b)$.

My main Approach:

I defined two functions $f(x)= \sin x $, and $g(x) = \cos x$. Then since both the functions were differentiable in the interval $(a,b)$, I applied the mean value theorem to $f$ and $g$ to get:

$(\sin(a) - \sin(b))/(a-b) = \cos(c)$ for some $c$ in $(a,b)$. And,

$(\cos(b) - \cos(a))/(a-b) = \sin(d)$ for some $d$ in $(a,b)$.

Now if $c$ was equal to $d$, the problem would have been super easy, however since it wasn't, I tried to get some kind of relationship between $c$ and $d$ by solving for $a-b$ on both equations and equating the results, but that didn't end well.

Question:

How to solve the above stated problem?

Thanks.

Answer 1

Hint:

Consider the function $$f(x)=\sin x-\sin b-\frac{\sin a-\sin b}{\cos a-\cos b}\,(\cos x-\cos b).$$ What are $f(a)$ and $f(b)$?