Question

if (tan theta - cot theta)=5 find the value of (tan^3 theta - cot^3 theta)(cot^2 theta - tan^2 theta)

Answer 1

$\textbf{Given:}$

$\bf\;tan\theta-cot\theta=5$

$\textbf{To find:}$

$(tan^3\theta-cot^3\theta)(cot^2\theta-tan^2\theta)$

$tan\theta-cot\theta=5$

$\text{Squaring on both sides, we get}$

$(tan\theta-cot\theta)^2=25$

$tan^2\theta+cot^2\theta-2\;tan\theta\,cot\theta=25$

$tan^2\theta+cot^2\theta-2(1)=25$

$tan^2\theta+cot^2\theta=25+2$

$\implies\bf\;tan^2\theta+cot^2\theta=27$

$\text{Add 2 on both sides, we get}$

$\implies\;tan^2\theta+cot^2\theta+2=29$

$\implies(tan\theta+cot\theta)^2=29$

$\implies\bf\,tan\theta+cot\theta=\sqrt{29}$

$\text{Now,}$

$(tan^3\theta-cot^3\theta)(cot^2\theta-tan^2\theta)$

$=(tan\theta-cot\theta)(tan^2\theta+tan\theta\,cot\theta+cot^2\theta)\;(cot\theta-tan\theta)(cot\theta+tan\theta)$

$=(tan\theta-cot\theta)(tan^2\theta+cot^2\theta+1)\;(cot\theta-tan\theta)(cot\theta+tan\theta)$

$=-(tan\theta-cot\theta)(tan^2\theta+cot^2\theta+1)\;(tan\theta-cot\theta)(cot\theta+tan\theta)$

$=-(5)(27+1)\;(5)(\sqrt{29})$

$=-(25)(28)(\sqrt{29})$

$=-700\sqrt{29}$

$\therefore\;\textbf{The value of}\bf\;(tan^3\theta-cot^3\theta)(cot^2\theta-tan^2\theta)\;\textbf{is}\bf\,-700\sqrt{29}$

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