Question

Tan3A- Tan2A-TanA=TanA*Tan2A*Tan3A

Answer 1

basically u need to use these formulas
\(\Large\rm \begin{align} \color{blue }{tan2\theta =\dfrac{2tan\theta}{1-tan^2\theta}\\~\\
tan3\theta =\dfrac{3tan\theta-tan^3\theta}{1-3tan^2\theta}\\~\\
}\end{align}\)

\(\large\tt \begin{align} \color{black }{tan3A- tan2A-tanA\\~\\
=\dfrac{3tanA-tan^3A}{1-3tan^2A}+\dfrac{-2tanA}{1-tan^2A}-tanA\\~\\
=\dfrac{3tanA-tan^3A}{1-3tan^2A}+ \dfrac{-2tanA}{1-tan^2A}-\dfrac{tanA(1-tan^2A)}{1-tan^2A}\\~\\
=\dfrac{3tanA-tan^3A}{1-3tan^2A}+ \dfrac{-2tanA-tanA(1-tan^2A)}{1-tan^2A}\\~\\
=\dfrac{3tanA-tan^3A}{1-3tan^2A}+ \dfrac{-3tanA+tan^3A}{1-tan^2A}\\~\\
=\dfrac{3tanA-tan^3A}{1-3tan^2A}- \dfrac{3tan-tan^3A}{1-tan^2A}\\~\\
=\dfrac{3tanA-tan^3A}{1-3tan^2A}- \dfrac{(3tan-tan^3A)}{1-tan^2A}\\~\\
=(3tanA-tan^3A)(\dfrac{1}{1-3tan^2A}- \dfrac{1}{1-tan^2A})\\~\\
=(3tanA-tan^3A)(\dfrac{1-tan^2A-(1-3tan^2A)}{(1-3tan^2A)(1-tan^2A)})\\~\\
=(3tanA-tan^3A)(\dfrac{2tan^2A}{(1-3tan^2A)(1-tan^2A)})\\~\\
=\dfrac{(3tanA-tan^3A)}{(1-3tan^2A)}\times \dfrac{2tanA}{(1-tan^2A)}\times tanA\\~\\~\\
\rm \Large =tan3A\times tan2A\times tanA\\~\\


}\end{align}\)