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Question
\[\frac{x^2 \cos\frac{\pi}{4}}{\sin x}\]
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Solution
\[\frac{x^2 \cos \frac{\pi}{4}}{\sin x} = x^2 \cos \frac{\pi}{4} cosec x\]
\[\text{ Let } u = x^2 ; v = \cos \frac{\pi}{4}; w = cosec x\]
\[\text{ Then }, u' = 2x; v' = 0; w' = - cosec x \cot x\]
\[\text{ Using the product rule }:\]
\[\frac{d}{dx}\left( uvw \right) = u'vw + uv'w + uvw'\]
\[\frac{d}{dx}\left( x^2 \cos \frac{\pi}{4} cosec x \right) = 2x \cos \frac{\pi}{4}cosec x + x^2 . 0 . cosec x + x^2 \cos \frac{\pi}{4}\left( - \cosec x \cot x \right)\]
\[ = \cos \frac{\pi}{4}\left( 2x cosec x - x^2 cosec x \cot x \right)\]
\[ = \cos \frac{\pi}{4}\left( \frac{2x}{\sin x} - x^2 \frac{\cot x}{\sin x} \right)\]
\[\]
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