er 4 sections 4.1, 4.2 & 4.4) test #4\nwhich expression, if any, is equivalent to: \\( \\frac { 2 \\tan…

er 4 sections 4.1, 4.2 & 4.4) test #4\nwhich expression, if any, is equivalent to: \\( \\frac { 2 \\tan \\left( \\frac { \\pi } { 8 } \\right) } { 1 - \\tan ^ { 2 } \\left( \\frac { \\pi } { 8 } \\right) } \\).\n\\( \\cot \\left( \\frac { 5 \\pi } { 4 } \\right) \\)\nnone of these.\n\\( \\cot \\left( \\frac { 4 \\pi } { 3 } \\right) \\)\n\\( \\tan \\left( \\frac { \\pi } { 3 } \\right) \\)

er 4 sections 4.1, 4.2 & 4.4) test #4\nwhich expression, if any, is equivalent to: \\( \\frac { 2 \\tan \\left( \\frac { \\pi } { 8 } \\right) } { 1 - \\tan ^ { 2 } \\left( \\frac { \\pi } { 8 } \\right) } \\).\n\\( \\cot \\left( \\frac { 5 \\pi } { 4 } \\right) \\)\nnone of these.\n\\( \\cot \\left( \\frac { 4 \\pi } { 3 } \\right) \\)\n\\( \\tan \\left( \\frac { \\pi } { 3 } \\right) \\)

Answer

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}))