Statement Using the addition formulae, find the exact values of: (a) cos15∘\cos 15^\circcos15∘ (b) tan105∘\tan 105^\circtan105∘ Solution (a) cos15∘=cos(45∘−30∘)=cos45∘cos30∘+sin45∘sin30∘=22⋅32+22⋅12=6+24≈0.9659\cos 15^\circ=\cos(45^\circ-30^\circ)=\cos45^\circ\cos30^\circ+\sin45^\circ\sin30^\circ=\dfrac{\sqrt2}{2}\cdot\dfrac{\sqrt3}{2}+\dfrac{\sqrt2}{2}\cdot\dfrac12=\dfrac{\sqrt6+\sqrt2}{4}\approx 0{.}9659cos15∘=cos(45∘−30∘)=cos45∘cos30∘+sin45∘sin30∘=22⋅23+22⋅21=46+2≈0.9659. (b) tan105∘=tan(60∘+45∘)=tan60∘+tan45∘1−tan60∘tan45∘=3+11−3=−(2+3)≈−3.7321\tan 105^\circ=\tan(60^\circ+45^\circ)=\dfrac{\tan60^\circ+\tan45^\circ}{1-\tan60^\circ\tan45^\circ}=\dfrac{\sqrt3+1}{1-\sqrt3}=-(2+\sqrt3)\approx -3{.}7321tan105∘=tan(60∘+45∘)=1−tan60∘tan45∘tan60∘+tan45∘=1−33+1=−(2+3)≈−3.7321. cos15∘=6+24;tan105∘=−(2+3)\boxed{\cos 15^\circ=\tfrac{\sqrt6+\sqrt2}{4};\qquad \tan 105^\circ=-(2+\sqrt3)}cos15∘=46+2;tan105∘=−(2+3)