Question : If A is an acute angle, the simplified form of
$\frac{\cos (\pi-A) \cdot \cot \left(\frac{\pi}{2}+A\right) \cos (-A)}{\tan (\pi+A) \tan \left(\frac{3 \pi}{2}+A\right) \sin (2 \pi-A)}$ is:
Option 1: $ \cos^2 A$
Option 2: $\sin A$
Option 3: $\sin^2 A$
Option 4: $\cos A$
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Correct Answer: $\cos A$
Solution :
$\frac{\cos (\pi-A) \cdot \cot \left(\frac{\pi}{2}+A\right) \cos (-A)}{\tan (\pi+A) \tan \left(\frac{3 \pi}{2}+A\right) \sin (2 \pi-A)}$
$=\frac{-\cos A \ \cdot (-\tan A) \ \cdot \cos A}{\tan A \ \cdot (-\cot A) (-\sin A)}$
$= \frac{\cos^2 A \ \cdot \tan A}{\sin A}$
$= \frac{\cos^2 A \ \cdot \frac{\sin A}{\cos A}}{\sin A}$
$= \cos A$
Hence, the correct answer is $\cos A$.
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