Question
Find the value of $\left(\sec ^2 \theta-1\right) \cdot \cot ^2 \theta$
✓
Answer
$\left(\sec ^2 \theta-1\right) \cdot \cot ^2 \theta$
Using the identity $\sec ^2 \theta-1=\tan ^2 \theta$, we get
$\left(\sec ^2 \theta-1\right) \cdot \cot ^2 \theta=\tan ^2 \theta \cdot \cot ^2 \theta$
$=\frac{\sin ^2 \theta}{\cos ^2 \theta} \cdot \frac{\cos ^2 \theta}{\sin ^2 \theta}=1$
$\left[\text { As } \tan \theta=\frac{\sin \theta}{\cos \theta} \text { and } \cot \theta=\frac{\cos \theta}{\sin \theta}\right]$
Using the identity $\sec ^2 \theta-1=\tan ^2 \theta$, we get
$\left(\sec ^2 \theta-1\right) \cdot \cot ^2 \theta=\tan ^2 \theta \cdot \cot ^2 \theta$
$=\frac{\sin ^2 \theta}{\cos ^2 \theta} \cdot \frac{\cos ^2 \theta}{\sin ^2 \theta}=1$
$\left[\text { As } \tan \theta=\frac{\sin \theta}{\cos \theta} \text { and } \cot \theta=\frac{\cos \theta}{\sin \theta}\right]$
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