The problem stated that
[tex]\tan 15=\frac{\sin 15}{\cos 15}=\frac{\sin\frac{30}{2}}{\cos\frac{30}{2}}[/tex]whereas sin 15 and cos 15 can be evaluated using half-angle formula. The half-angle formula for sine function is
[tex]\sin \frac{\theta}{2}=\sqrt[]{\frac{1-\cos \theta}{2}}[/tex]while the half-angle formula for cosine function is
[tex]\cos \frac{\theta}{2}=\sqrt[]{\frac{1+\cos \theta}{2}}[/tex]The theta value to evaluate the half-angle formula for tangent 15 is 30 degrees. We now proceed in calculating the value of the trigonometric function.
[tex]\begin{gathered} \sin \frac{30}{2}=\sqrt[]{\frac{1-\cos30}{2}} \\ \cos \frac{30}{2}=\sqrt[]{\frac{1+\cos30}{2}} \end{gathered}[/tex]The exact value of cosine 30 is √3/2, hence, the equation above now becomes
[tex]\begin{gathered} \sin \frac{30}{2}=\sqrt[]{\frac{1-\frac{\sqrt[]{3}}{2}}{2}} \\ \cos \frac{30}{2}=\sqrt[]{\frac{1+\frac{\sqrt[]{3}}{2}}{2}} \end{gathered}[/tex]Simplifying the equations above
[tex]\begin{gathered} \sin \frac{30}{2}=\sqrt[]{\frac{\frac{2-\sqrt[]{3}}{2}}{2}}=\sqrt[]{\frac{2-\sqrt[]{3}}{4}} \\ \cos \frac{30}{2}=\sqrt[]{\frac{\frac{2+\sqrt[]{3}}{2}}{2}=}\sqrt[]{\frac{2+\sqrt[]{3}}{4}} \end{gathered}[/tex]Since tangent theta is the ratio between the sin function and the cosine function, getting the ratio of the two equations above will result to
[tex]\frac{\sin\frac{30}{2}}{\cos\frac{30}{2}}=\tan 15=\frac{\sqrt[]{\frac{2-\sqrt[]{3}}{4}}}{\sqrt[]{\frac{2+\sqrt[]{3}}{4}}}[/tex]Simplifying,
[tex]\begin{gathered} \tan 15=\sqrt[]{\frac{2-\sqrt[]{3}}{4}}\cdot\sqrt[]{\frac{4}{2+\sqrt[]{3}}} \\ \tan 15=\frac{\sqrt[]{2-\sqrt[]{3}}}{\sqrt[]{2+\sqrt[]{3}}}=\sqrt[]{\frac{2-\sqrt[]{3}}{2+\sqrt[]{3}}} \end{gathered}[/tex]Where the equation above shows the exact value of tan 15.
