Question

(sec^2 theta-1)(1-cosec^2 theta)
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Answer 1

Question :

$\sf \Big[sec^{2} \theta - 1\Big]\Big[1 - cosec^{2} \theta\Big]$

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Solution :

$\sf \Big[sec^{2} \theta - 1\Big]\Big[1 - cosec^{2} \theta\Big]$

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We know that :

$\large \underline{\boxed{\bf{sec^{2} \theta - 1 = tan^{2} \theta }}}$

$\sf : \implies \Big[tan^{2} \theta\Big]\Big[1 - cosec^{2} \theta\Big]$

$\sf : \implies \Big[tan^{2} \theta\Big]\Big[- ( -1 + cosec^{2} \theta)\Big]$

$\sf : \implies \Big[tan^{2} \theta\Big]\Big[- (cosec^{2} \theta - 1)\Big]$

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We know that :

$\large \underline{\boxed{\bf{cosec^{2} \theta - 1 = cot^{2} \theta }}}$

$\sf : \implies \Big[tan^{2} \theta\Big]\Big[- cot^{2} \theta\Big]$

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We know that :

$\large \underline{\boxed{\bf{cot \theta = \dfrac{1}{tan \theta}}}}$

$\sf : \implies \Big[tan^{2} \theta\Big] \Big[- \dfrac{1}{tan^{2} \theta}\Big]$

$\sf : \implies \Big[\cancel{tan^{2} \theta}\Big] \Big[- \dfrac{1}{\cancel{tan^{2} \theta}} \Big]$

$\sf : \implies - 1$

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Hence,

$\sf \Big[sec^{2} \theta - 1\Big]\Big[1 - cosec^{2} \theta\Big] = - 1$

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Some Trigonometric identities :

• $\sf sin^{2} \theta + cos^{2} \theta = 1$
• $\sf sec^{2} \theta - tan^{2} \theta = 1$
• $\sf cosec^{2} \theta - cot^{2} \theta = 1$