$\tan\left(7\dfrac{1}{2}\right)^o=\tan\left(\dfrac{15}{2}\right)^o$
$=\dfrac{\sin\dfrac{15^o}{2}}{\cos\dfrac{15^o}{2}}$
$=\dfrac{2\sin\dfrac{15^o}{2}\times \sin\dfrac{15^o}{2}}{2\sin\dfrac{15^o}{2}\times \cos\dfrac{15^o}{2}}$
$=\dfrac{2\sin^2\dfrac{15^o}{2}}{\sin\left(2\times\dfrac{15^o}{2}\right)}$
$=\dfrac{1-\cos\left(2\times\dfrac{15^o}{2}\right)}{\sin 15^o}$
$=\dfrac{1-\cos 15^o}{\sin 15^o}$
Now,
$\cos 15^o=\cos(45^o-30^o)$
$=\cos 45^o\cos30^o+\sin 45^o\sin 30^o$
$=\dfrac{1}{\sqrt{2}}\times\dfrac{\sqrt{3}}{2}+\dfrac{1}{\sqrt{2}}\times\dfrac {1}{2}$
$=\dfrac{\sqrt{3}+1}{2\sqrt{2}}$
$\sin 15^o=\sin(45^o-30^o)$
$=\sin 45^o\cos30^o-\cos 45^o\sin 30^o$
$=\dfrac{1}{\sqrt{2}}\times\dfrac{\sqrt{3}}{2}-\dfrac{1}{\sqrt{2}}\times\dfrac {1}{2}$
$=\dfrac{\sqrt{3}-1}{2\sqrt{2}}$
$\tan\left(7\dfrac{1}{2}\right)^o=\dfrac{1-\dfrac{\sqrt{3}+1}{2\sqrt{2}}}{\dfrac{\sqrt{3}-1}{2\sqrt{2}}}$
$=\dfrac{2\sqrt{2}-(\sqrt{3}+1)}{\sqrt{3}-1}$