Simplify $ \left(1+\tan ^{2} \theta\right)(1-\sin \theta)(1+\sin \theta) $

To do:

We have to simplify \( \left(1+\tan ^{2} \theta\right)(1-\sin \theta)(1+\sin \theta) \).

Solution:

We know that,

$\sin^2 \theta+\cos ^{2} \theta=1$.....(i)

$\sec^2 \theta-\tan ^{2} \theta=1$.......(ii)

$\sec \theta \times \cos \theta=1$.......(iii)

Therefore,

$\left(1+\tan ^{2} \theta\right)(1-\sin \theta)(1+\sin \theta)=\left(1+\tan ^{2} \theta\right)(1^2-\sin^2 \theta)$          [Since $(a-b)(a+b)=a^2-b^2$]

$=\left(\sec ^{2} \theta\right)(\cos^2 \theta)$           [From (i) and (ii)]

$=(\sec \theta\times cos \theta)^2$                  

$=1^2$                          [From (iii)]

$=1$ 

Therefore,

\( \left(1+\tan ^{2} \theta\right)(1-\sin \theta)(1+\sin \theta)=1 \).

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Updated on: 10-Oct-2022

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