If${\ tan}{\alpha, an{\beta, an{\gamma\ }}}$are the roots of the equation $x^{3} - px^{2} - r = 0,$ then the value of $\left( 1 + an^{2}\alpha ight)\left( 1 + an^{2}\beta ight)\left( 1 + an^{2}\gamma ight)$ is equal to

Question

If${\ tan}{\alpha,	an{\beta,	an{\gamma\ }}}$are the roots of the equation $x^{3} - px^{2} - r = 0,$ then the value of $\left( 1 + 	an^{2}\alpha ight)\left( 1 + 	an^{2}\beta ight)\left( 1 + 	an^{2}\gamma ight)$ is equal to

Dr. Abubacker · Accepted Answer

(b)
From the given equations we have $\sum_{}^{}{	an{\alpha = p}}$
$\Sigma	an\alpha	an{\beta = 0\ and	an{\alpha	an{\beta	an{\gamma = r}}}}$
$	herefore\ \left( 1 + 	an^{2}\alpha ight)\left( 1 + 	an^{2}\beta ight)\left( 1 + 	an^{2}\gamma ight)$
$= 1 + \ \Sigma	an^{2}{\alpha + \Sigma	an^{2}{\alpha	an^{2}{\beta + 	an^{2}{\alpha	an^{2}{\beta	an^{2}\gamma}}}}}$
$= 1 + \left( \Sigma	an\alpha ight)^{2} - 2\Sigma	an{\alpha	an{\beta + \ \left( \Sigma	an{\alpha	an\beta} ight)^{2}}} - 2	an{\alpha	an{\beta	an{\gamma\ \Sigma	an{\alpha + 	an^{2}{\alpha	an^{2}{\beta	an^{2}\gamma}}}}}}$
$= 1 + p^{2} - 2pr + r^{2} = 1 + (p - r)^{2}$

Q. If${\ tan}{\alpha,\tan{\beta,\tan{\gamma\ }}}$are the roots of the equation $x^{3} - px^{2} - r = 0,$ then the value of $\left( 1 + \tan^{2}\alpha \right)\left( 1 + \tan^{2}\beta \right)\left( 1 + \tan^{2}\gamma \right)$ is equal to

Explanation

Reviewed by Dr. Abubacker

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