1.

Write the value of tan ( \(\frac{\pi}{8}\)):

Answer»

We have 

tan (\(\frac{\pi}{8}\)) = tan ( \(\frac{1}{2},\frac{\pi}{4}\) ) 

Using the half angle formulae, 

\(tan(\frac{\alpha}{8})=\frac{sin\alpha}{1+cos\alpha}\)

\(tan(\frac{\pi}{8})=\frac{sin(\frac{\pi}{4})}{1+cos(\frac{\pi}{4})}\) \(=\frac{\frac{1}{\sqrt{2}}}{1+\frac{1}{\sqrt{2}}}=\frac{1}{1+\sqrt{2}}\)

Given, \(secβ=\frac{-5}{3},\frac{\pi}{2}<β<\pi\)

\(tan^2β=1-sec^2β\) \(=1-\frac{25}{9}=\frac{-16}{9}\) 

\(tanβ=\frac{-4}{3}\)                 (∵ \(\frac{p}{2}<β<p\)

And \(cot\alpha=\frac{1}{2},\)              \(\frac{\pi}{2}<\alpha<\frac{3\pi}{2}\)



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