Question
Prove the following.Simplify $(1+\tan^2\theta)(1-\sin\theta)(1+\sin\theta)$
✓
Answer
$(1+\tan^2\theta)(1-\sin\theta)(1+\sin\theta)=(1+\tan^2\theta)(1-\sin^2\theta)$ $[\because(\text{a}-\text{b})(\text{a}+\text{b})=\text{a}^2-\text{b}^2]$
$=\sec^2\theta\cdot\cos^2\theta$
$[\because1+\tan^2\theta=\sec^2\theta\text{ and }\cos^2\theta+\sin^2\theta=1]$
$=\frac{1}{\cos^2\theta}\cdot\cos^2\theta=1\ \Big[\because\sec\theta=\frac{1}{\cos\theta}\Big]$
$=\sec^2\theta\cdot\cos^2\theta$
$[\because1+\tan^2\theta=\sec^2\theta\text{ and }\cos^2\theta+\sin^2\theta=1]$
$=\frac{1}{\cos^2\theta}\cdot\cos^2\theta=1\ \Big[\because\sec\theta=\frac{1}{\cos\theta}\Big]$
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