Chapter 7 - Trigonometric Identities and Equations - 7.4 Double-Angle and Half-Angle Identities - 7.4 Exercises - Page 694: 70

$\cot 82.5=\tan7.5$

$\sqrt {\dfrac {1+\cos 165}{1-\cos 165}} =\sqrt {\dfrac {1+\cos \left( 2\times \dfrac {165}{2}\right) }{1-\cos \left( 2\times \dfrac {165}{2}\right) }}=\sqrt {\dfrac {1+\left( \cos ^{2}\dfrac {165}{2}-\sin ^{2}\dfrac {165}{2}\right) }{1-\left( \cos ^{2}\dfrac {165}{2}-\sin ^{2}\dfrac {165}{2}\right) }}=\sqrt {\dfrac {2\cos ^{2}\dfrac {165}{2}}{2\sin ^{2}\dfrac {165}{2}}}=\cot \dfrac {165}{2}=\cot 82.5=\tan7.5$