Question

express tan4A in terms of tan A

Answer 1

Dear student
$$\begin{gathered} \tan 4\mathrm{A} \\ =\tan \left(2\times 2\mathrm{A}\right) \\ =\frac{2\tan 2\mathrm{A}}{1-{\tan }^{2}2\mathrm{A}}\left[\because \tan 2\mathrm{x}=\frac{2\mathrm{tanx}}{1-{\tan }^2\mathrm{x}}\right] \\ =\frac{2\times {\displaystyle \frac{2\mathrm{tanA}}{1-{\tan }^2\mathrm{A}}}}{1-{\left({\displaystyle \frac{2\mathrm{tanA}}{1-{\tan }^2\mathrm{A}}}\right)}^2} \\ =\frac{{\displaystyle \frac{4\mathrm{tanA}}{1-{\tan }^2\mathrm{A}}}}{{\displaystyle \frac{{\left(1-{\tan }^2\mathrm{A}\right)}^2-4{\tan }^2\mathrm{A}}{{\left(1-{\tan }^2\mathrm{A}\right)}^2}}} \\ =\frac{4\mathrm{tanA}\left(1-{\tan }^2\mathrm{A}\right)}{1+{\tan }^4\mathrm{A}-2{\tan }^2\mathrm{A}-4{\tan }^2\mathrm{A}} \\ =\frac{4\mathrm{tanA}\left(1-{\tan }^2\mathrm{A}\right)}{1+{\tan }^4\mathrm{A}-6{\tan }^2\mathrm{A}} \end{gathered}$$
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