Question

er 4 sections 4.1, 4.2 & 4.4) test #4
which expression, if any, is equivalent to: \\( \frac { 2 \tan \left( \frac { \pi } { 8 } \
ight) } { 1 - \tan ^ { 2 } \left( \frac { \pi } { 8 } \
ight) } \\).
\\( \cot \left( \frac { 5 \pi } { 4 } \
ight) \\)
none of these.
\\( \cot \left( \frac { 4 \pi } { 3 } \
ight) \\)
\\( \tan \left( \frac { \pi } { 3 } \
ight) \\)

Explanation:

Step1: Use double - angle formula for tangent

The double - angle formula for tangent is \(\tan(2\alpha)=\frac{2\tan\alpha}{1 - \tan^{2}\alpha}\).
Let \(\alpha=\frac{\pi}{8}\), then \(\frac{2\tan(\frac{\pi}{8})}{1-\tan^{2}(\frac{\pi}{8})}=\tan(2\times\frac{\pi}{8})=\tan(\frac{\pi}{4}) = 1\).

Step2: Evaluate each option

• For \(\cot(\frac{5\pi}{4})\):

Using the identity \(\cot\theta=\frac{1}{\tan\theta}\), \(\cot(\frac{5\pi}{4})=\frac{1}{\tan(\frac{5\pi}{4})}\). Since \(\tan(\frac{5\pi}{4})=\tan(\pi+\frac{\pi}{4})=\tan(\frac{\pi}{4}) = 1\), then \(\cot(\frac{5\pi}{4}) = 1\).

• For \(\cot(\frac{4\pi}{3})\):

\(\cot(\frac{4\pi}{3})=\frac{1}{\tan(\frac{4\pi}{3})}\), \(\tan(\frac{4\pi}{3})=\tan(\pi+\frac{\pi}{3})=\tan(\frac{\pi}{3})=\sqrt{3}\), so \(\cot(\frac{4\pi}{3})=\frac{1}{\sqrt{3}}\).

• For \(\tan(\frac{\pi}{3})\): \(\tan(\frac{\pi}{3})=\sqrt{3}\).

Answer:

\(\cot(\frac{5\pi}{4})\)