\({\left( {\frac{{1 - tan\theta }}{{1 - \cot \theta }}} \right)^2} + 1 = ?\)
sec 2θ
Step 1 — Simplify the inner fraction: write \(\tan\theta=\tfrac{\sin\theta}{\cos\theta}\) and \(\cot\theta=\tfrac{\cos\theta}{\sin\theta}\).
\[\frac{1-\tan\theta}{1-\cot\theta}=\frac{\frac{\cos\theta-\sin\theta}{\cos\theta}}{\frac{\sin\theta-\cos\theta}{\sin\theta}}=\frac{\sin\theta}{\cos\theta}\cdot\frac{\cos\theta-\sin\theta}{-(\cos\theta-\sin\theta)}=-\tan\theta\]
Step 2 — Square and add 1:
\[(-\tan\theta)^2+1=\tan^2\theta+1=\sec^2\theta\]
Therefore the value is \(\sec^2\theta\).
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