Prove the following trigonometric identities:$ \left(\sec ^{2} \theta-1\right)\left(\operatorname{cosec}^{2} \theta-1\right)=1 $

To do:

We have to prove that \( \left(\sec ^{2} \theta-1\right)\left(\operatorname{cosec}^{2} \theta-1\right)=1 \).

Solution: We know that,

$\sec ^{2} A-tan ^{2} A=1$.......(i)

$\operatorname{cosec}^{2} A-cot ^{2} A=1$.......(ii)

$ \tan ^2 A\times\cot ^2 A=1$.......(iii)

Therefore,

$\left(\sec ^{2} \theta-1\right)\left(\operatorname{cosec}^{2} \theta-1\right)=(\tan ^{2} \theta)(\cot ^{2} \theta)$       (From (i) and (ii))

$=\tan ^{2} \theta \times \cot ^{2} \theta$

$=1$              (From (iii))

Hence proved. 

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Updated on: 10-Oct-2022

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