This problem recently came up in an FP3 (old P6) textbook, and none of my 7-strong further maths class have been able to solve it

Prove by induction that:

$\displaystyle

\sum_{r=1}^n\ \tan(r\theta)\tan((r+1)\theta)=\tan((n+1)\theta) \cot(\theta)-n-1

$

Thanks for any help you can give; this problem has really started to irritate us!