Question:

Find \((\tan 15^o + \sqrt{3})^2\) without a calculator.

Answer:

It is not easy to obtain \(\tan 15^o\) without a calculator. In this case, you may want to come up with angles which can easily convert into specific numbers, such as 30, 45, 60, 90, etc. In fact, 15 = 45 – 30, so \(\tan 15^o = \tan(45^o – 30^o)\). Thus, we can use the addition theorem of tangent.

\[ \tan(45^o – 30^o) = \frac{\tan 45^o – \tan 30^o}{1 + \tan 45^o \tan 30^o} \]

If we use typical triangles, we know \(\tan 45^o = 1\) and \(\tan 30^o = 1/\sqrt{3}\). Therefore,

\[ \tan 15^o = \frac{1 – 1/\sqrt{3}}{1 + 1 \times 1/\sqrt{3}} \]

Multiply \(\sqrt{3}\) by both numerator and denominator.

\[ = \frac{1 – 1/\sqrt{3}}{1 + 1/\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} \]

\[ = \frac{\sqrt{3}-1}{\sqrt{3}+1} \]

\[ = \frac{\sqrt{3}-1}{\sqrt{3}+1} \times \frac{\sqrt{3}-1}{\sqrt{3}-1} \]

\[ = \frac{4 – 2\sqrt{3}}{2} = 2 – \sqrt{3}\]

Hence, we have

\[ (\tan 15^o + \sqrt{3})^2 = 2^2 = 4 \]

What do you think?