What is the value of (1 + tan2 θ) (1 − sin θ) (1 + sin θ)?

We have,

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

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

We know that,

$\sec ^{2} \theta-\tan ^{2} \theta=1$

$\Rightarrow \sec ^{2} \theta=1+\tan ^{2} \theta$

$\sin ^{2} \theta+\cos ^{2} \theta=1$

$\Rightarrow \cos ^{2} \theta=1-\sin ^{2} \theta$

Therefore,

$\left(1+\tan ^{2} \theta\right)(1-\sin \theta)(1+\sin \theta)=\sec ^{2} \theta \times \cos ^{2} \theta$

$=\frac{1}{\cos ^{2} \theta} \times \cos ^{2} \theta$

$=1$